INTERNATIONAL GONGRESS F ¯ X∙(A)=Hom(G_(m),A_( bar(F)))\mathbb{X} \bullet(A)=\operatorname{Hom}\left(\mathbb{G}_{m}, A_{\bar{F}}\right)X∙(A)=Hom⁡(Gm,AF¯) the group of its cocharacters.
For a prime â„“ â„“\ellâ„“, let Λ Î› Lambda\LambdaΛ be F , Z , Q F â„“ , Z â„“ , Q â„“ F_(â„“),Z_(â„“),Q_(â„“)\mathbb{F}_{\ell}, \mathbb{Z}_{\ell}, \mathbb{Q}_{\ell}Fâ„“,Zâ„“,Qâ„“ or a finite (flat) extension of such rings. It will serve as the coefficient ring of our sheaf theory.

1. FROM CLASSICAL TO GEOMETRIC LANGLANDS CORRESPONDENCE

In this section, we review some developments of the geometric Langlands theory inspired from the classical theory, with another important source of inspiration from quantum physics. The basic idea is categorification/geometrization, which is a process of replacing set-theoretic statements with categorical analogues
(1.1) Numbers Vector spaces Categories 2-Categories . (1.1)  Numbers  →  Vector spaces  →  Categories  →  2-Categories  → ⋯ .  {:(1.1)" Numbers "rarr" Vector spaces "rarr" Categories "rarr" 2-Categories "rarr cdots". ":}\begin{equation*} \text { Numbers } \rightarrow \text { Vector spaces } \rightarrow \text { Categories } \rightarrow \text { 2-Categories } \rightarrow \cdots \text {. } \tag{1.1} \end{equation*}(1.1) Numbers → Vector spaces → Categories → 2-Categories →⋯. 
We illustrate this process by some important examples.

1.1. The geometric Satake

The starting point of the Langlands program is (Langlands' interpretation of) the Satake isomorphism, in which the Langlands dual group appears mysteriously. Similarly, the starting point of the geometric Langlands theory is the geometric Satake equivalence, which is a tensor equivalence between the category of perverse sheaves on the (spherical) local Hecke stack of a connected reductive group and the category of finite-dimensional algebraic representations of its dual group. This is a vast generalization of the classical Satake isomorphism (via the sheaf-to-function dictionary), and arguably gives a conceptual explanation why the Langlands dual group (in fact, the C C CCC-group) should appear in the Langlands correspondence.
We follow [83, SECT. 1.1] for notations and conventions regarding dual groups. Let G G GGG denote a connected reductive group over a field F F FFF. Let ( G ^ , B ^ , T ^ , e ^ ) ( G ^ , B ^ , T ^ , e ^ ) ( hat(G), hat(B), hat(T), hat(e))(\hat{G}, \hat{B}, \hat{T}, \hat{e})(G^,B^,T^,e^) be a pinned Langlands dual group of G G GGG over Z Z Z\mathbb{Z}Z. There is a finite Galois extension F ~ / F F ~ / F tilde(F)//F\tilde{F} / FF~/F, and a natural injective map ξ : Γ F ~ / F Aut ( G ^ , B ^ , T ^ , e ^ ) ξ : Γ F ~ / F ⊂ Aut ⁡ ( G ^ , B ^ , T ^ , e ^ ) xi:Gamma_( tilde(F)//F)sub Aut( hat(G), hat(B), hat(T), hat(e))\xi: \Gamma_{\tilde{F} / F} \subset \operatorname{Aut}(\hat{G}, \hat{B}, \hat{T}, \hat{e})ξ:ΓF~/F⊂Aut⁡(G^,B^,T^,e^), induced by the action of Γ F Γ F Gamma_(F)\Gamma_{F}ΓF on the root datum of G G GGG. Let L G = G ^ Γ F ~ / F L G = G ^ ⋊ Γ F ~ / F ^(L)G= hat(G)><|Gamma_( tilde(F)//F){ }^{L} G=\hat{G} \rtimes \Gamma_{\tilde{F} / F}LG=G^⋊ΓF~/F denote the usual L L LLL-group of G G GGG, and c G = G ^ ( G m × Γ F ~ / F ) c G = G ^ ⋊ G m × Γ F ~ / F ^(c)G= hat(G)><|(G_(m)xxGamma_( tilde(F)//F)){ }^{c} G=\hat{G} \rtimes\left(\mathbb{G}_{m} \times \Gamma_{\tilde{F} / F}\right)cG=G^⋊(Gm×ΓF~/F) the group defined in [83], which is isomorphic to the C C CCC-group of G G GGG introduced by Buzzard-Gee. We write d : c G G m × Γ F ~ / F d : c G → G m × Γ F ~ / F d:^(c)G rarrG_(m)xxGamma_( tilde(F)//F)d:{ }^{c} G \rightarrow \mathbb{G}_{m} \times \Gamma_{\tilde{F} / F}d:cG→Gm×ΓF~/F for the projection with the kernel G ^ G ^ hat(G)\hat{G}G^.
Now let F F FFF be a nonarchimedean local field with O O O\mathcal{O}O being its ring of integers and k = F q k = F q k=F_(q)k=\mathbb{F}_{q}k=Fq its residue field. That is, F F FFF is a finite extension of Q p Q p Q_(p)\mathbb{Q}_{p}Qp or is isomorphic to F q ( ( ϖ ) ) F q ( ( Ï– ) ) F_(q)((Ï–))\mathbb{F}_{q}((\varpi))Fq((Ï–)). Let σ σ sigma\sigmaσ be the geometric q q qqq-Frobenius of k k kkk. Assume that G G GGG can be extended to a connected reductive group over O O O\mathcal{O}O (such G G GGG is called unramified), and we fix such an extension to have G ( O ) G ( F ) G ( O ) ⊂ G ( F ) G(O)sub G(F)G(\mathcal{O}) \subset G(F)G(O)⊂G(F), usually called a hyperspecial subgroup of G ( F ) G ( F ) G(F)G(F)G(F). With a basis of open neighborhoods of the unit given by finite-index subgroups of G ( O ) G ( O ) G(O)G(\mathcal{O})G(O), the group G ( F ) G ( F ) G(F)G(F)G(F) is a locally compact topological group. The classical spherical Hecke algebra is the space of compactly supported G ( O ) G ( O ) G(O)G(\mathcal{O})G(O)-biinvariant C C C\mathbb{C}C-valued functions on G ( F ) G ( F ) G(F)G(F)G(F), equipped with the convolution product
(1.2) ( f g ) ( x ) = G ( F ) f ( y ) g ( y 1 x ) d y (1.2) ( f ∗ g ) ( x ) = ∫ G ( F )   f ( y ) g y − 1 x d y {:(1.2)(f**g)(x)=int_(G(F))f(y)g(y^(-1)x)dy:}\begin{equation*} (f * g)(x)=\int_{G(F)} f(y) g\left(y^{-1} x\right) d y \tag{1.2} \end{equation*}(1.2)(f∗g)(x)=∫G(F)f(y)g(y−1x)dy
where d y d y dyd ydy is the Haar measure on G ( F ) G ( F ) G(F)G(F)G(F) such that G ( O ) G ( O ) G(O)G(\mathcal{O})G(O) has volume 1 . Note that if both f f fff and g g ggg are Z Z Z\mathbb{Z}Z-valued, so is f g f ∗ g f**gf * gf∗g. Therefore, the subset H G ( O ) c l H G ( O ) c l H_(G(O))^(cl)H_{G(\mathcal{O})}^{\mathrm{cl}}HG(O)cl of Z Z Z\mathbb{Z}Z-valued functions forms a Z Z Z\mathbb{Z}Z-algebra. 1 1 ^(1){ }^{1}1
On the dual side, under the unramifiedness assumption, Γ F ~ / F Γ F ~ / F Gamma_( tilde(F)//F)\Gamma_{\tilde{F} / F}ΓF~/F is a finite cyclic group generated by σ σ sigma\sigmaσ. Note that G ^ G ^ hat(G)\hat{G}G^ acts on c G | d = ( q , σ ) c G d = ( q , σ ) ^(c)G|_(d=(q,sigma))\left.{ }^{c} G\right|_{d=(q, \sigma)}cG|d=(q,σ), the fiber of d d ddd at ( q , σ ) G m × Γ F ~ / F ( q , σ ) ∈ G m × Γ F ~ / F (q,sigma)inG_(m)xxGamma_( tilde(F)//F)(q, \sigma) \in \mathbb{G}_{m} \times \Gamma_{\tilde{F} / F}(q,σ)∈Gm×ΓF~/F, by conjugation. Then the classical Satake isomorphism establishes a canonical isomorphism of Z [ q 1 ] Z q − 1 Z[q^(-1)]\mathbb{Z}\left[q^{-1}\right]Z[q−1]-algebras
(1.3) S a t c l : Z [ q 1 ] [ c G | d = ( q , σ ) ] G ^ H G ( O ) c l Z [ q 1 ] (1.3) S a t c l : Z q − 1 c G d = ( q , σ ) G ^ ≅ H G ( O ) c l ⊗ Z q − 1 {:(1.3)Sat^(cl):Z[q^(-1)][^(c)G|_(d=(q,sigma))]^( hat(G))~=H_(G(O))^(cl)oxZ[q^(-1)]:}\begin{equation*} \mathrm{Sat}^{\mathrm{cl}}: \mathbb{Z}\left[q^{-1}\right]\left[\left.{ }^{c} G\right|_{d=(q, \sigma)}\right]^{\hat{G}} \cong H_{G(\mathcal{O})}^{\mathrm{cl}} \otimes \mathbb{Z}\left[q^{-1}\right] \tag{1.3} \end{equation*}(1.3)Satcl:Z[q−1][cG|d=(q,σ)]G^≅HG(O)cl⊗Z[q−1]
Remark 1.1.1. In fact, as explained in [83], there is a Satake isomorphism over Z Z Z\mathbb{Z}Z (without inverting q q qqq ), in which the C C CCC-group c G c G ^(c)G{ }^{c} GcG is replaced by certain affine monoid containing it as the group of invertible elements. On the other hand, if we extend the base ring from Z [ q 1 ] Z q − 1 Z[q^(-1)]\mathbb{Z}\left[q^{-1}\right]Z[q−1] to Z [ q ± 1 2 ] Z q ± 1 2 Z[q^(+-(1)/(2))]\mathbb{Z}\left[q^{ \pm \frac{1}{2}}\right]Z[q±12], one can rewrite (1.3) as an isomorphism
(1.4) Z [ q ± 1 2 ] [ G ^ σ ] G ^ H G ( O ) c l Z [ q ± 1 2 ] (1.4) Z q ± 1 2 [ G ^ σ ] G ^ ≅ H G ( O ) c l ⊗ Z q ± 1 2 {:(1.4)Z[q^(+-(1)/(2))][ hat(G)sigma]^( hat(G))~=H_(G(O))^(cl)oxZ[q^(+-(1)/(2))]:}\begin{equation*} \mathbb{Z}\left[q^{ \pm \frac{1}{2}}\right][\hat{G} \sigma]^{\hat{G}} \cong H_{G(\mathcal{O})}^{\mathrm{cl}} \otimes \mathbb{Z}\left[q^{ \pm \frac{1}{2}}\right] \tag{1.4} \end{equation*}(1.4)Z[q±12][G^σ]G^≅HG(O)cl⊗Z[q±12]
where G ^ G ^ hat(G)\hat{G}G^ acts on G ^ σ L G G ^ σ ⊂ L G hat(G)sigma sub^(L)G\hat{G} \sigma \subset{ }^{L} GG^σ⊂LG by the usual conjugation (e.g., see [83] for the discussion). This is the more traditional formulation of the Satake isomorphism, which is slightly noncanonical, but suffices for many applications.
In the geometric theory, where instead of thinking G ( F ) G ( F ) G(F)G(F)G(F) as a topological group and considering the space of G ( O ) G ( O ) G(O)G(\mathcal{O})G(O)-biinvariant compactly supported functions on it, one regards
1 Here ( ) c l ( − ) c l (-)^(cl)(-)^{\mathrm{cl}}(−)cl stands for the classical Hecke algebra, as opposed to the derived Hecke algebra mentioned in (2.2).
G ( F ) G ( F ) G(F)G(F)G(F) as a certain algebro-geometric object and studies the category of G ( O ) G ( O ) G(O)G(\mathcal{O})G(O)-biequivariant sheaves on it. In the rest of the section, we allow F F FFF to be slightly more general. Namely, we assume that F F FFF is a local field complete with respect to a discrete valuation, with ring of integers O O O\mathcal{O}O and a perfect residue field k k kkk of characteristic p > 0 . 2 p > 0 . 2 p > 0.^(2)p>0 .{ }^{2}p>0.2 Let ϖ O Ï– ∈ O Ï–inO\varpi \in \mathcal{O}ϖ∈O be a uniformizer.
We work in the realm of perfect algebraic geometry. Recall that a k k kkk-algebra R R RRR is called perfect if the Frobenius endomorphism R R , r r p R → R , r ↦ r p R rarr R,r|->r^(p)R \rightarrow R, r \mapsto r^{p}R→R,r↦rp, is a bijection. Let A f f k p f A f f k p f Aff_(k)^(pf)\mathbf{A f f}_{k}^{\mathrm{pf}}Affkpf denote the category of perfect k k kkk-algebras. By a perfect presheaf (or, more generally, a perfect prestack), we mean a functor from A f f k p f A f f k p f Aff_(k)^(pf)\mathbf{A f f}_{k}^{\mathrm{pf}}Affkpf to the category Sets of sets (or, more generally, a functor from A f f k p f A f f k p f Aff_(k)^(pf)\mathbf{A f f}{ }_{k}^{\mathrm{pf}}Affkpf to the ∞ oo\infty∞-category S p c S p c Spc\mathbf{S p c}Spc of spaces). Many constructions in usual algebraic geometry work in this setting. For example, one can endow Aff k p f k p f _(k)^(pf){ }_{k}^{\mathrm{pf}}kpf with Zariski, étale, or fpqc topology as usual and talk about corresponding sheaves and stacks. One can then define perfect schemes, perfect algebraic spaces, perfect algebraic stacks, etc., as sheaves (stacks) with certain properties. It turns out that the category of perfect schemes/algebraic spaces defined this way is equivalent to the category of perfect schemes/algebraic spaces in the usual sense. Some foundations of perfect algebraic geometry can be found in [78, APPENDIX A], [13] and [64, SECT. A.1].
For a perfect k k kkk-algebra R R RRR, let W O ( R ) W O ( R ) W_(O)(R)W_{\mathcal{O}}(R)WO(R) denote the ring of Witt vectors in R R RRR with coefficient in O O O\mathcal{O}O. If char F = F = F=F=F= char k k kkk, then W O ( R ) R [ [ ϖ ] ] W O ( R ) ≃ R [ [ Ï– ] ] W_(O)(R)≃R[[Ï–]]W_{\mathcal{O}}(R) \simeq R[[\varpi]]WO(R)≃R[[Ï–]]. If char F F ≠ F!=F \neqF≠ char k k kkk, see [78, SECT. 0.5]. If R = k ¯ R = k ¯ R= bar(k)R=\bar{k}R=k¯, we denote W O ( k ¯ ) W O ( k ¯ ) W_(O)( bar(k))W_{\mathcal{O}}(\bar{k})WO(k¯) by O F ˘ O F ˘ O_(F^(˘))\mathcal{O}_{\breve{F}}OF˘ and W O ( k ¯ ) [ 1 / ϖ ] W O ( k ¯ ) [ 1 / Ï– ] W_(O)( bar(k))[1//Ï–]W_{\mathcal{O}}(\bar{k})[1 / \varpi]WO(k¯)[1/Ï–] by F ˘ F ˘ F^(˘)\breve{F}F˘. We write D R = Spec W O ( R ) D R = Spec ⁡ W O ( R ) D_(R)=Spec W_(O)(R)D_{R}=\operatorname{Spec} W_{\mathcal{O}}(R)DR=Spec⁡WO(R) and D R = Spec W O ( R ) [ 1 / ϖ ] D R ∗ = Spec ⁡ W O ( R ) [ 1 / Ï– ] D_(R)^(**)=Spec W_(O)(R)[1//Ï–]D_{R}^{*}=\operatorname{Spec} W_{\mathcal{O}}(R)[1 / \varpi]DR∗=Spec⁡WO(R)[1/Ï–] which are thought as a family of (punctured) discs parameterized by Spec R Spec ⁡ R Spec R\operatorname{Spec} RSpec⁡R.
We denote by L + G L + G L^(+)GL^{+} GL+G (resp. L G L G LGL GLG ) the jet group (resp. loop group) of G G GGG. As presheaves on Aff k pf k pf  _(k)^("pf "){ }_{k}^{\text {pf }}kpf ,
L + G ( R ) = G ( W O ( R ) ) , L G ( R ) = G ( W O ( R ) [ 1 / ϖ ] ) L + G ( R ) = G W O ( R ) , L G ( R ) = G W O ( R ) [ 1 / Ï– ] L^(+)G(R)=G(W_(O)(R)),quad LG(R)=G(W_(O)(R)[1//Ï–])L^{+} G(R)=G\left(W_{\mathcal{O}}(R)\right), \quad L G(R)=G\left(W_{\mathcal{O}}(R)[1 / \varpi]\right)L+G(R)=G(WO(R)),LG(R)=G(WO(R)[1/Ï–])
Note that L + G ( k ) = G ( O ) L + G ( k ) = G ( O ) L^(+)G(k)=G(O)L^{+} G(k)=G(\mathcal{O})L+G(k)=G(O) and L G ( k ) = G ( F ) L G ( k ) = G ( F ) LG(k)=G(F)L G(k)=G(F)LG(k)=G(F). Let
H k G := L + G L G / L + G H k G := L + G ∖ L G / L + G Hk_(G):=L^(+)G\\LG//L^(+)G\mathrm{Hk}_{G}:=L^{+} G \backslash L G / L^{+} GHkG:=L+G∖LG/L+G
be the étale stack quotient of L G L G LGL GLG by the left and right L + G L + G L^(+)GL^{+} GL+G-action, sometimes called the (spherical) local Hecke stack of G G GGG. As a perfect prestack, it sends R R RRR to triples ( E 1 , E 2 , β ) E 1 , E 2 , β (E_(1),E_(2),beta)\left(\mathcal{E}_{1}, \mathcal{E}_{2}, \beta\right)(E1,E2,β), where ε 1 , ε 2 ε 1 , ε 2 epsi_(1),epsi_(2)\varepsilon_{1}, \varepsilon_{2}ε1,ε2 are two G G GGG-torsors on D R D R D_(R)D_{R}DR, and β : ε 1 | D R ε 2 | D R β : ε 1 D R ∗ ≃ ε 2 D R ∗ beta:epsi_(1)|_(D_(R)^(**))≃epsi_(2)|_(D_(R)^(**))\beta:\left.\left.\varepsilon_{1}\right|_{D_{R}^{*}} \simeq \varepsilon_{2}\right|_{D_{R}^{*}}β:ε1|DR∗≃ε2|DR∗ is an isomorphism.
For p â„“ ≠ p â„“!=p\ell \neq pℓ≠p, the modern developments of higher category theory allow one to define the ∞ oo\infty∞-category of étale F F â„“ F_(â„“)\mathbb{F}_{\ell}Fâ„“-sheaves on any prestack (e.g., see [35]). In particular, for Λ = F , Z , Q Λ = F â„“ , Z â„“ , Q â„“ Lambda=F_(â„“),Z_(â„“),Q_(â„“)\Lambda=\mathbb{F}_{\ell}, \mathbb{Z}_{\ell}, \mathbb{Q}_{\ell}Λ=Fâ„“,Zâ„“,Qâ„“ (or finite extension of these rings), it is possible to define the ∞ oo\infty∞ category Shv ( H k G , Λ ) Shv ⁡ H k G , Λ Shv(Hk_(G),Lambda)\operatorname{Shv}\left(\mathrm{Hk}_{G}, \Lambda\right)Shv⁡(HkG,Λ) of Λ Î› Lambda\LambdaΛ-sheaves on H k G H k G Hk_(G)\mathrm{Hk}_{G}HkG, which is the categorical analogue of the space of G ( O ) G ( O ) G(O)G(\mathcal{O})G(O)-biinvariant functions on G ( F ) G ( F ) G(F)G(F)G(F). But without knowing some geometric properties of H k G H k G Hk_(G)\mathrm{Hk}_{G}HkG, very little can be said about Shv ( H k G , Λ ) Shv ⁡ H k G , Λ Shv(Hk_(G),Lambda)\operatorname{Shv}\left(\mathrm{Hk}_{G}, \Lambda\right)Shv⁡(HkG,Λ). The crucial geometric input is the following theorem.
Theorem 1.1.2. Let Gr G := L G / L + G Gr G := L G / L + G Gr_(G):=LG//L^(+)G\operatorname{Gr}_{G}:=L G / L^{+} GGrG:=LG/L+G be the étale quotient of L G L G LGL GLG by the (right) L + G L + G L^(+)GL^{+} GL+G action, which admits the left L + G L + G L^(+)GL^{+} GL+G-action. Then Gr G Gr G Gr_(G)\operatorname{Gr}_{G}GrG can be written as an inductive limit of L + G L + G L^(+)GL^{+} GL+G-stable subfunctors lim X i → lim X i rarr"lim"X_(i)\xrightarrow{\lim } X_{i}→limXi, with each X i X i X_(i)X_{i}Xi being a perfect projective variety and X i X i + 1 X i → X i + 1 X_(i)rarrX_(i+1)X_{i} \rightarrow X_{i+1}Xi→Xi+1 being a closed embedding.
The space Gr G Gr G Gr_(G)\operatorname{Gr}_{G}GrG is usually called the affine Grassmannian of G G GGG. See [4,23] for the equal characteristic case and [ 13 , 78 ] [ 13 , 78 ] [13,78][13,78][13,78] for the mixed characteristic case, and see [77,80] for examples of closed subvarieties in G r G G r G Gr_(G)\mathrm{Gr}_{G}GrG. The theorem allows one to define the category of constructible and perverse sheaves on H k G H k G Hk_(G)\mathrm{Hk}_{G}HkG, and to formulate the geometric Satake, as we discuss now.
First, the (left) quotient by L + G L + G L^(+)GL^{+} GL+G-action induces a map G r G H k G G r G → H k G Gr_(G)rarrHk_(G)\mathrm{Gr}_{G} \rightarrow \mathrm{Hk}_{G}GrG→HkG. Roughly speaking, a sheaf on H k G H k G Hk_(G)\mathrm{Hk}_{G}HkG is perverse (resp. constructible) if its pullback to G r G G r G Gr_(G)\mathrm{Gr}_{G}GrG comes from a perverse (resp. constructible) sheaf on some X i X i X_(i)X_{i}Xi. Then inside S h v ( H k G , Λ ) S h v H k G , Λ Shv(Hk_(G),Lambda)\mathbf{S h v}\left(\mathrm{Hk}_{G}, \Lambda\right)Shv(HkG,Λ) we have the categories Perv ( H k G , Λ ) Shv c ( H k G , Λ ) Perv ⁡ H k G , Λ ⊂ Shv c ⁡ H k G , Λ Perv(Hk_(G),Lambda)subShv_(c)(Hk_(G),Lambda)\operatorname{Perv}\left(\mathrm{Hk}_{G}, \Lambda\right) \subset \operatorname{Shv}_{c}\left(\mathrm{Hk}_{G}, \Lambda\right)Perv⁡(HkG,Λ)⊂Shvc⁡(HkG,Λ) of perverse and constructible sheaves on H k G H k G Hk_(G)\mathrm{Hk}_{G}HkG. They can be regarded as categorical analogues of the space of G ( O ) G ( O ) G(O)G(\mathcal{O})G(O)-biinvariant compactly supported functions on G ( F ) G ( F ) G(F)G(F)G(F). In addition, Perv ( H k G , Λ ) Perv ⁡ H k G , Λ Perv(Hk_(G),Lambda)\operatorname{Perv}\left(\mathrm{Hk}_{G}, \Lambda\right)Perv⁡(HkG,Λ) is an abelian category, semisimple if Λ Î› Lambda\LambdaΛ is a field of characteristic zero, called the Satake category of G G GGG. For simplicity, we assume that Λ Î› Lambda\LambdaΛ is a field in the sequel. 3 3 ^(3){ }^{3}3
There is also a categorical analogue of the convolution product (1.2). Namely, there is the convolution diagram
H k G × H k G p r L + G L G × L + G L G / L + G m H k G H k G × H k G ← p r L + G ∖ L G × L + G L G / L + G → m H k G Hk_(G)xxHk_(G)larr^(pr)L^(+)G\\LG xx^(L^(+)G)LG//L^(+)Grarr"m"Hk_(G)\mathrm{Hk}_{G} \times \mathrm{Hk}_{G} \stackrel{\mathrm{pr}}{\leftarrow} L^{+} G \backslash L G \times{ }^{L^{+} G} L G / L^{+} G \xrightarrow{m} \mathrm{Hk}_{G}HkG×HkG←prL+G∖LG×L+GLG/L+G→mHkG
and the convolution of two sheaves A , B S h v ( H k G , Λ ) A , B ∈ S h v H k G , Λ A,BinShv(Hk_(G),Lambda)\mathscr{A}, \mathscr{B} \in \mathbf{S h v}\left(\mathrm{Hk}_{G}, \Lambda\right)A,B∈Shv(HkG,Λ) is defined as
(1.5) A B := m ! pr ( A B ) (1.5) A ⋆ B := m ! pr ∗ ⁡ ( A ⊠ B ) {:(1.5)A***B:=m!pr^(**)(A⊠B):}\begin{equation*} \mathcal{A} \star \mathscr{B}:=m!\operatorname{pr}^{*}(\mathscr{A} \boxtimes \mathscr{B}) \tag{1.5} \end{equation*}(1.5)A⋆B:=m!pr∗⁡(A⊠B)
This convolution product makes S h v ( H k G , Λ ) S h v H k G , Λ Shv(Hk_(G),Lambda)\mathbf{S h v}\left(\mathrm{Hk}_{G}, \Lambda\right)Shv(HkG,Λ) into a monoidal ∞ oo\infty∞-category containing Perv ( H k G , Λ ) Shv c ( H k G , Λ ) Perv ⁡ H k G , Λ ⊂ Shv c ⁡ H k G , Λ Perv(Hk_(G),Lambda)subShv_(c)(Hk_(G),Lambda)\operatorname{Perv}\left(\mathrm{Hk}_{G}, \Lambda\right) \subset \operatorname{Shv}_{c}\left(\mathrm{Hk}_{G}, \Lambda\right)Perv⁡(HkG,Λ)⊂Shvc⁡(HkG,Λ) as monoidal subcategories.
Remark 1.1.3. The above construction of the Satake category as a monoidal category is essentially equivalent to the more traditional approach, in which the Satake category is defined as the category of L + G L + G L^(+)GL^{+} GL+G-equivariant perverse sheaves on Gr G Gr G Gr_(G)\operatorname{Gr}_{G}GrG (e.g., see [80] for an exposition).
Let Coh ( B G ^ Λ ) ρ Coh ⁡ B G ^ Λ ρ Coh (B hat(G)_(Lambda))^(rho)\operatorname{Coh}\left(\mathbb{B} \hat{G}_{\Lambda}\right)^{\rho}Coh⁡(BG^Λ)ρ denote the abelian monoidal category of coherent sheaves on the classifying stack B G ^ Λ B G ^ Λ B hat(G)_(Lambda)\mathbb{B} \hat{G}_{\Lambda}BG^Λ over Λ 4 Λ 4 Lambda^(4)\Lambda^{4}Λ4, which is equivalent to the category of algebraic representations of G ^ G ^ hat(G)\hat{G}G^ on finite dimensional Λ Î› Lambda\LambdaΛ-vector spaces. This following theorem is usually known as the geometric Satake equivalence.
Theorem 1.1.4. There is a canonical equivalence of monoidal abelian categories
Sat G : Coh ( B G ^ Λ ) Perv ( H k G k ¯ , Λ ) Sat G : Coh ⁡ B G ^ Λ â—¯ ≅ Perv ⁡ H k G ⊗ k ¯ , Λ Sat_(G):Coh (B hat(G)_(Lambda))^(â—¯)~=Perv(Hk_(G)ox( bar(k)),Lambda)\operatorname{Sat}_{G}: \operatorname{Coh}\left(\mathbb{B} \hat{G}_{\Lambda}\right)^{\bigcirc} \cong \operatorname{Perv}\left(\mathrm{Hk}_{G} \otimes \bar{k}, \Lambda\right)SatG:Coh⁡(BG^Λ)◯≅Perv⁡(HkG⊗k¯,Λ)
3 The formulation for Λ = Z Λ = Z â„“ Lambda=Z_(â„“)\Lambda=\mathbb{Z}_{\ell}Λ=Zâ„“ is slightly more complicated, as the right-hand side of (1.5) may not be perverse when A A A\mathscr{A}A and B B B\mathscr{B}B are perverse.
4 In the dual group side, we always work in the realm of usual algebraic geometry, so B G ^ B G ^ B hat(G)\mathbb{B} \hat{G}BG^ is an Artin stack in the usual sense.
Geometric Satake is really one of the cornerstones of the geometric Langlands program, and has found numerous applications to representation theory, mathematical physics, and (arithmetic) algebraic geometry. When F = k ( ( ϖ ) ) F = k ( ( Ï– ) ) F=k((Ï–))F=k((\varpi))F=k((Ï–)), this theorem grew out of works of Lusztig, Ginzburg, Beilinson-Drinfeld and Mirković-Vilonen (cf. [5,51,53]). In mixed characteristic, it was proved in [ 69 , 78 ] [ 69 , 78 ] [69,78][69,78][69,78], with the equal characteristic case as an input, and in [25] by mimicking the strategy in equal characteristic. We conclude this subsection with a few remarks.
Remark 1.1.5. (1) As mentioned before, the geometric Satake can be regarded as the conceptual definition of the Langlands dual group G ^ G ^ hat(G)\hat{G}G^ of G G GGG, namely as the Tannakian group of the Tannakian category Perv ( H k G k ¯ , Λ ) Perv ⁡ H k G ⊗ k ¯ , Λ Perv(Hk_(G)ox( bar(k)),Lambda)\operatorname{Perv}\left(\mathrm{Hk}_{G} \otimes \bar{k}, \Lambda\right)Perv⁡(HkG⊗k¯,Λ). In addition, as explained in [72, 76], the group G ^ G ^ hat(G)\hat{G}G^ is canonically equipped with a pinning ( B ^ , T ^ , e ^ ) ( B ^ , T ^ , e ^ ) ( hat(B), hat(T), hat(e))(\hat{B}, \hat{T}, \hat{e})(B^,T^,e^). In the rest of the article, by the pinned Langlands dual group ( G ^ , B ^ , T ^ , e ^ ) G ^ , B ^ , T ^ , e ^ ) hat(G), hat(B), hat(T), hat(e))\hat{G}, \hat{B}, \hat{T}, \hat{e})G^,B^,T^,e^) of G G GGG, we mean the quadruple defined by the geometric Satake.
(2) For arithmetic applications, one needs to understand the Γ k Γ k Gamma_(k)\Gamma_{k}Γk-action on Perv ( H k G k ¯ , Λ ) Perv ⁡ H k G ⊗ k ¯ , Λ Perv(Hk_(G)ox( bar(k)),Lambda)\operatorname{Perv}\left(\mathrm{Hk}_{G} \otimes \bar{k}, \Lambda\right)Perv⁡(HkG⊗k¯,Λ) in terms of the dual group side. It turns out that such an action is induced by an action of Γ k Γ k Gamma_(k)\Gamma_{k}Γk on G ^ G ^ hat(G)\hat{G}G^, preserving ( B ^ , T ^ ) ( B ^ , T ^ ) ( hat(B), hat(T))(\hat{B}, \hat{T})(B^,T^) but not e ^ e ^ hat(e)\hat{e}e^. See [76, 80], or [77] from the motivic point of view. This leads to the appearance of the group c G c G ^(c)G{ }^{c} GcG. See [ 76 , 80 , 83 ] [ 76 , 80 , 83 ] [76,80,83][76,80,83][76,80,83] for detailed discussions.
(3) There is also the derived Satake equivalence [11], describing Shv c ( H k G k ¯ , Λ ) Shv c ⁡ H k G ⊗ k ¯ , Λ Shv_(c)(Hk_(G)ox( bar(k)),Lambda)\operatorname{Shv}_{c}\left(\mathrm{Hk}_{G} \otimes \bar{k}, \Lambda\right)Shvc⁡(HkG⊗k¯,Λ) in terms of the dual group, at least when Λ Î› Lambda\LambdaΛ is a field of characteristic zero. We mention that the category in the dual side is not the derived category of coherent sheaves on B G ^ Λ B G ^ Λ B hat(G)_(Lambda)\mathbb{B} \hat{G}_{\Lambda}BG^Λ.
(4) In fact, for many applications, it is important to have a family version of the geometric Satake. For a (nonempty) finite set S S SSS, there is a local Hecke stack H k G , D S H k G , D S Hk_(G,D^(S))\mathrm{Hk}_{G, D^{S}}HkG,DS over D S D S D^(S)D^{S}DS, the self-product of the disc D = Spec O D = Spec ⁡ O D=Spec OD=\operatorname{Spec} \mathcal{O}D=Spec⁡O over S S SSS, which, roughly speaking, classifies quadruples ( { x s } s S , E , E , β ) x s s ∈ S , E , E ′ , β ({x_(s)}_(s in S),E,E^('),beta)\left(\left\{x_{s}\right\}_{s \in S}, \mathcal{E}, \mathcal{E}^{\prime}, \beta\right)({xs}s∈S,E,E′,β), where { x s } s S x s s ∈ S {x_(s)}_(s in S)\left\{x_{s}\right\}_{s \in S}{xs}s∈S is an S S SSS-tuple of points of D , E D , E D,ED, \mathcal{E}D,E and E E ′ E^(')\mathcal{E}^{\prime}E′ are two G G GGG-torsors on D D DDD, and β β beta\betaβ is an isomorphism between E E E\mathcal{E}E and E E ′ E^(')\mathcal{E}^{\prime}E′ on D s { x s } D − ⋃ s   x s D-uuu_(s){x_(s)}D-\bigcup_{s}\left\{x_{s}\right\}D−⋃s{xs}. In equal characteristic, one can regard D D DDD as the formal disc at a k k kkk-point of an algebraic curve X X XXX over k k kkk and H k G , D S H k G , D S Hk_(G,D^(S))\mathrm{Hk}_{G, D^{S}}HkG,DS is the restriction along D S X S D S → X S D^(S)rarrX^(S)D^{S} \rightarrow X^{S}DS→XS of the stack
H k G , X S = ( L + G ) X S ( L G ) X S / ( L + G ) X S H k G , X S = L + G X S ∖ ( L G ) X S / L + G X S Hk_(G,X^(S))=(L^(+)G)_(X^(S))\\(LG)_(X^(S))//(L^(+)G)_(X^(S))\mathrm{Hk}_{G, X^{S}}=\left(L^{+} G\right)_{X^{S}} \backslash(L G)_{X^{S}} /\left(L^{+} G\right)_{X^{S}}HkG,XS=(L+G)XS∖(LG)XS/(L+G)XS
where ( L G ) X S ( L G ) X S (LG)_(X^(S))(L G)_{X^{S}}(LG)XS and ( L + G ) X S L + G X S (L^(+)G)_(X^(S))\left(L^{+} G\right)_{X^{S}}(L+G)XS are family versions of L G L G LGL GLG and L + G L + G L^(+)GL^{+} GL+G over X S X S X^(S)X^{S}XS (e.g., see [80, SECT. 3.1] for precise definitions). In mixed characteristic, the stack H k G , D S H k G , D S Hk_(G,D^(S))\mathrm{Hk}_{G, D^{S}}HkG,DS (and in fact D S D S D^(S)D^{S}DS itself) does not live in the world of (perfect) algebraic geometry, but rather in the world of perfectoid analytic geometry as developed by Scholze (see [25,59]). In both cases, one can consider certain category Perv U L A ( H k G , D S k ¯ , Λ ) U L A H k G , D S ⊗ k ¯ , Λ ^(ULA)(Hk_(G,D^(S))ox( bar(k)),Lambda)^{\mathrm{ULA}}\left(\mathrm{Hk}_{G, D^{S}} \otimes \bar{k}, \Lambda\right)ULA(HkG,DS⊗k¯,Λ) of (ULA) perverse sheaves on H k G , D s k ¯ H k G , D s ⊗ k ¯ Hk_(G,D^(s))ox bar(k)\mathrm{Hk}_{G, D^{s}} \otimes \bar{k}HkG,Ds⊗k¯. In addition, for a map S S S → S ′ S rarrS^(')S \rightarrow S^{\prime}S→S′ of finite sets, restriction along H k G , D S H k G , D s H k G , D S ′ → H k G , D s Hk_(G,D^(S^(')))rarrHk_(G,D^(s))\mathrm{Hk}_{G, D^{S^{\prime}}} \rightarrow \mathrm{Hk}_{G, D^{s}}HkG,DS′→HkG,Ds gives a functor Perv U L A ( H k G , D S k ¯ , Λ ) Perv U L A ( H k G , D S k ¯ , Λ ) . 5 U L A H k G , D S ⊗ k ¯ , Λ → Perv U L A ⁡ H k G , D S ′ ⊗ k ¯ , Λ . 5 ^(ULA)(Hk_(G,D^(S))ox( bar(k)),Lambda)rarrPerv^(ULA)(Hk_(G,D^(S^(')))ox( bar(k)),Lambda).^(5){ }^{\mathrm{ULA}}\left(\mathrm{Hk}_{G, D^{S}} \otimes \bar{k}, \Lambda\right) \rightarrow \operatorname{Perv}^{\mathrm{ULA}}\left(\mathrm{Hk}_{G, D^{S^{\prime}}} \otimes \bar{k}, \Lambda\right) .{ }^{5}ULA(HkG,DS⊗k¯,Λ)→PervULA⁡(HkG,DS′⊗k¯,Λ).5 We refer to the above mentioned references for details.
On the other hand, let G ^ S G ^ S hat(G)^(S)\hat{G}^{S}G^S be the S S SSS-power self-product of G ^ G ^ hat(G)\hat{G}G^ over Λ Î› Lambda\LambdaΛ. Then for S S S → S ′ S rarrS^(')S \rightarrow S^{\prime}S→S′, the restriction along B G ^ S B G ^ S B G ^ S ′ → B G ^ S B hat(G)^(S^('))rarrB hat(G)^(S)\mathbb{B} \hat{G}^{S^{\prime}} \rightarrow \mathbb{B} \hat{G}^{S}BG^S′→BG^S gives a functor Coh ( B G ^ Λ S ) Coh ( B G ^ Λ S ) Coh ⁡ B G ^ Λ S â—¯ → Coh ⁡ B G ^ Λ S ′ â—¯ Coh (B hat(G)_(Lambda)^(S))^(â—¯)rarr Coh (B hat(G)_(Lambda)^(S^(')))^(â—¯)\operatorname{Coh}\left(\mathbb{B} \hat{G}_{\Lambda}^{S}\right)^{\bigcirc} \rightarrow \operatorname{Coh}\left(\mathbb{B} \hat{G}_{\Lambda}^{S^{\prime}}\right)^{\bigcirc}Coh⁡(BG^ΛS)◯→Coh⁡(BG^ΛS′)â—¯. Now a family version of the geometric Satake gives a system of functors
(1.6) Sat S : Coh ( B G ^ Λ S ) Perv U L A ( H k G , D s k ¯ , Λ ) (1.6) Sat S : Coh ⁡ B G ^ Λ S â—¯ →  Perv  U L A H k G , D s ⊗ k ¯ , Λ {:(1.6)Sat_(S):Coh (B hat(G)_(Lambda)^(S))^(â—¯)rarr" Perv "^(ULA)(Hk_(G,D^(s))ox( bar(k)),Lambda):}\begin{equation*} \operatorname{Sat}_{S}: \operatorname{Coh}\left(\mathbb{B} \hat{G}_{\Lambda}^{S}\right)^{\bigcirc} \rightarrow \text { Perv }^{\mathrm{ULA}}\left(\mathrm{Hk}_{G, D^{s}} \otimes \bar{k}, \Lambda\right) \tag{1.6} \end{equation*}(1.6)SatS:Coh⁡(BG^ΛS)◯→ Perv ULA(HkG,Ds⊗k¯,Λ)
compatible with restriction functors on both sides induced by maps between finite sets (see [ 28 , 80 ] [ 28 , 80 ] [28,80][28,80][28,80] ).
(5) For applications, it is important to have the geometric Satake in different sheaftheoretic contents over different versions of local Hecke stacks. Besides the above mentioned ones, we also mention a D D DDD-module version [5], and an arithmetic D D DDD-module version [66].

1.2. Tamely ramified local geometric Langlands correspondence

We first recall the classical theory. Assume that F F FFF is a finite extension of Q p Q p Q_(p)\mathbb{Q}_{p}Qp or is isomorphic to F q ( ( ϖ ) ) F q ( ( Ï– ) ) F_(q)((Ï–))\mathbb{F}_{q}((\varpi))Fq((Ï–)), and for simplicity assume that G G GGG extends to a connected reductive group over O O O\mathcal{O}O. (In fact, results in the subsection hold in appropriate forms for quasi-split groups that are split over a tamely ramified extension of F F FFF.) In addition, we fix a pinning ( B , T , e ) ( B , T , e ) (B,T,e)(B, T, e)(B,T,e) of G G GGG over O O O\mathcal{O}O.
The classical local Langlands program aims to classify (smooth) irreducible representations of G ( F ) G ( F ) G(F)G(F)G(F) (over C C C\mathbb{C}C ) in terms of Galois representations. From this point of view, the Satake isomorphism (1.3) gives a classification of irreducible unramified representations of G ( F ) G ( F ) G(F)G(F)G(F), i.e., those that have nonzero vectors fixed by G ( O ) G ( O ) G(O)G(\mathcal{O})G(O), as such representations are in one-to-one correspondence with simple modules of H G ( O ) c l C H G ( O ) c l ⊗ C H_(G(O))^(cl)oxCH_{G(\mathcal{O})}^{\mathrm{cl}} \otimes \mathbb{C}HG(O)cl⊗C, which via the Satake isomorphism (1.3) are parameterized by semisimple G ^ G ^ hat(G)\hat{G}G^-conjugacy classes in c G c G ^(c)G{ }^{c} GcG. (For an irreducible unramified representation π Ï€ pi\piÏ€, the corresponding G ^ G ^ hat(G)\hat{G}G^-conjugacy class in c G c G ^(c)G{ }^{c} GcG is usually called the Satake parameter of π Ï€ pi\piÏ€.)
The next important class of irreducible representations are those that have nonzero vectors fixed by an Iwahori subgroup G ( F ) G ( F ) G(F)G(F)G(F). For example, under the reduction mod ϖ Ï– Ï–\varpiÏ– map G ( O ) G ( k ) G ( O ) → G ( k ) G(O)rarr G(k)G(\mathcal{O}) \rightarrow G(k)G(O)→G(k), the preimage I I III of B ( k ) G ( k ) B ( k ) ⊂ G ( k ) B(k)sub G(k)B(k) \subset G(k)B(k)⊂G(k) is an Iwahori subgroup of G ( F ) G ( F ) G(F)G(F)G(F). As in the unramified case, the Z Z Z\mathbb{Z}Z-valued I I III-biinvariant functions form a Z Z Z\mathbb{Z}Z-algebra H I c l H I c l H_(I)^(cl)H_{I}^{\mathrm{cl}}HIcl with multiplication given by convolution (1.2) (with the Haar measure d g d g dgd gdg chosen so that the volume of I I III is one), and the set of irreducible representations of G ( F ) G ( F ) G(F)G(F)G(F) that have nonzero I I III-fixed vectors are in bijection with the set of simple ( H I c l C ) H I c l ⊗ C (H_(I)^(cl)oxC)\left(H_{I}^{\mathrm{cl}} \otimes \mathbb{C}\right)(HIcl⊗C)-modules. Just as the Satake isomorphism, Kazhdan-Lusztig gave a description of H I c l C H I c l ⊗ C H_(I)^(cl)oxCH_{I}^{\mathrm{cl}} \otimes \mathbb{C}HIcl⊗C in terms of geometric objects associated to G ^ G ^ hat(G)\hat{G}G^.
Let U ^ B ^ U ^ ⊂ B ^ hat(U)sub hat(B)\hat{U} \subset \hat{B}U^⊂B^ denote the unipotent radical of B ^ B ^ hat(B)\hat{B}B^. The natural morphism U ^ / B ^ G ^ / G ^ U ^ / B ^ → G ^ / G ^ hat(U)// hat(B)rarr hat(G)// hat(G)\hat{U} / \hat{B} \rightarrow \hat{G} / \hat{G}U^/B^→G^/G^ is usually called the Springer resolution. Let
S G ^ u n i p = ( U ^ / B ^ ) × G ^ / G ^ ( U ^ / B ^ ) S G ^ u n i p = ( U ^ / B ^ ) × G ^ / G ^ ( U ^ / B ^ ) S_( hat(G))^(unip)=( hat(U)// hat(B))xx_( hat(G)// hat(G))( hat(U)// hat(B))S_{\hat{G}}^{\mathrm{unip}}=(\hat{U} / \hat{B}) \times_{\hat{G} / \hat{G}}(\hat{U} / \hat{B})SG^unip=(U^/B^)×G^/G^(U^/B^)

and therefore on S G ^ , C unip S G ^ , C unip  S_( hat(G),C)^("unip ")S_{\hat{G}, \mathbb{C}}^{\text {unip }}SG^,Cunip , by identifying U ^ C U ^ C hat(U)_(C)\hat{U}_{\mathbb{C}}U^C with its Lie algebra via the exponential map. Then
one can form the quotient stack S G ^ , C unip / G m , C S G ^ , C unip  / G m , C S_( hat(G),C)^("unip ")//G_(m,C)S_{\hat{G}, \mathbb{C}}^{\text {unip }} / \mathbb{G}_{m, \mathbb{C}}SG^,Cunip /Gm,C. In the sequel, for an Artin stack X X XXX (of finite presentation) over C C C\mathbb{C}C, we let K ( X ) K ( X ) K(X)K(X)K(X) denote the K K KKK-group of the ( ( ∞ (oo(\infty(∞-)category of coherent sheaves on X X XXX.
Kazhdan-Lusztig [36] constructed (under the assumption that G G GGG is split with connected center) a canonical isomorphism (after choosing a square root of q q sqrtq\sqrt{q}q of q q qqq )
(1.7) K ( S G ^ , C unip / G m , C ) K ( B G m , C ) C H I c l C (1.7) K S G ^ , C unip  / G m , C ⊗ K B G m , C C ≅ H I c l ⊗ C {:(1.7)K(S_( hat(G),C)^("unip ")//G_(m,C))ox_(K(BG_(m,C)))C~=H_(I)^(cl)oxC:}\begin{equation*} K\left(S_{\hat{G}, \mathbb{C}}^{\text {unip }} / \mathbb{G}_{m, \mathbb{C}}\right) \otimes_{K\left(\mathbb{B} \mathbb{G}_{m, \mathbb{C}}\right)} \mathbb{C} \cong H_{I}^{\mathrm{cl}} \otimes \mathbb{C} \tag{1.7} \end{equation*}(1.7)K(SG^,Cunip /Gm,C)⊗K(BGm,C)C≅HIcl⊗C
where the map K ( B G m , C ) C K B G m , C → C K(BG_(m,C))rarrCK\left(\mathbb{B} \mathbb{G}_{m, \mathbb{C}}\right) \rightarrow \mathbb{C}K(BGm,C)→C sends the class corresponding to the tautological representation of G m , C G m , C G_(m,C)\mathbb{G}_{m, \mathbb{C}}Gm,C to q q sqrtq\sqrt{q}q. In addition, the isomorphism induces the Bernstein isomorphism
(1.8) K ( B G ^ C ) C Z ( H I c l C ) (1.8) K B G ^ C ⊗ C ≅ Z H I c l ⊗ C {:(1.8)K(B hat(G)_(C))oxC~=Z(H_(I)^(cl)oxC):}\begin{equation*} K\left(\mathbb{B} \hat{G}_{\mathbb{C}}\right) \otimes \mathbb{C} \cong Z\left(H_{I}^{\mathrm{cl}} \otimes \mathbb{C}\right) \tag{1.8} \end{equation*}(1.8)K(BG^C)⊗C≅Z(HIcl⊗C)
where Z ( H I c l C ) Z H I c l ⊗ C Z(H_(I)^(cl)oxC)Z\left(H_{I}^{\mathrm{cl}} \otimes \mathbb{C}\right)Z(HIcl⊗C) is the center of H I c l C H I c l ⊗ C H_(I)^(cl)oxCH_{I}^{\mathrm{cl}} \otimes \mathbb{C}HIcl⊗C, and the map K ( B G ^ C ) K ( S G ^ , C unip / G m , C ) K B G ^ C → K S G ^ , C unip  / G m , C K(B hat(G)_(C))rarr K(S_( hat(G),C)^("unip ")//G_(m,C))K\left(\mathbb{B} \hat{G}_{\mathbb{C}}\right) \rightarrow K\left(S_{\hat{G}, \mathbb{C}}^{\text {unip }} / \mathbb{G}_{m, \mathbb{C}}\right)K(BG^C)→K(SG^,Cunip /Gm,C) is induced by the natural projection S G ^ unip / G m B G ^ S G ^ unip  / G m → B G ^ S_( hat(G))^("unip ")//G_(m)rarrB hat(G)S_{\hat{G}}^{\text {unip }} / \mathbb{G}_{m} \rightarrow \mathbb{B} \hat{G}SG^unip /Gm→BG^.
Remark 1.2.1. It would be interesting to give a description of the Z Z Z\mathbb{Z}Z-algebra H I c l H I c l H_(I)^(cl)H_{I}^{\mathrm{cl}}HIcl in terms of the geometry involving G ^ G ^ hat(G)\hat{G}G^, which would generalize the integral Satake isomorphism from [83].
It turns out that the Kazhdan-Lusztig isomorphism (1.7) also admits a categorification, usually known as the Bezrukavnikov equivalence, which gives two realizations of the affine Hecke category. Again, when switching to the geometric theory, we allow F F FFF to be a little bit more general as in Section 1.1. We also assume that G G GGG extends to a connected reductive group over O O O\mathcal{O}O and fix a pinning of G G GGG over O O O\mathcal{O}O. Let L + G G k L + G → G k L^(+)G rarrG_(k)L^{+} G \rightarrow G_{k}L+G→Gk be the natural "reduction mod ϖ Ï– Ï–\varpiÏ– " map, and let Iw L + G ⊂ L + G subL^(+)G\subset L^{+} G⊂L+G be the preimage of B k G k B k ⊂ G k B_(k)subG_(k)B_{k} \subset G_{k}Bk⊂Gk. This is the geometric analogue of I I III. Then as in the unramified case discussed in Section 1.1, one can define the Iwahori local Hecke stack H k I w = I w L G / I w H k I w = I w ∖ L G / I w Hk_(Iw)=Iw\\LG//Iw\mathrm{Hk}_{\mathrm{Iw}}=\mathrm{Iw} \backslash L G / \mathrm{Iw}HkIw=Iw∖LG/Iw and the monoidal categories Shv c ( H k I w k ¯ , Λ ) S h v ( H k I w k ¯ , Λ ) Shv c ⁡ H k I w ⊗ k ¯ , Λ ⊂ S h v H k I w ⊗ k ¯ , Λ Shv_(c)(Hk_(Iw)ox( bar(k)),Lambda)subShv(Hk_(Iw)ox( bar(k)),Lambda)\operatorname{Shv}_{c}\left(\mathrm{Hk}_{\mathrm{Iw}} \otimes \bar{k}, \Lambda\right) \subset \mathbf{S h v}\left(\mathrm{Hk}_{\mathrm{Iw}} \otimes \bar{k}, \Lambda\right)Shvc⁡(HkIw⊗k¯,Λ)⊂Shv(HkIw⊗k¯,Λ). The category S h v v c ( H k I w k ¯ , Λ ) S h v v c H k I w ⊗ k ¯ , Λ Shvv_(c)(Hk_(Iw)ox( bar(k)),Lambda)\mathbf{S h v} v_{c}\left(\mathrm{Hk}_{\mathrm{Iw}} \otimes \bar{k}, \Lambda\right)Shvvc(HkIw⊗k¯,Λ) can be thought as a categorical analogue of H I c l H I c l H_(I)^(cl)H_{I}^{\mathrm{cl}}HIcl, usually called the affine Hecke category.
Recall that we let F ˘ = W O ( k ¯ ) [ 1 / ϖ ] F ˘ = W O ( k ¯ ) [ 1 / Ï– ] F^(˘)=W_(O)( bar(k))[1//Ï–]\breve{F}=W_{\mathcal{O}}(\bar{k})[1 / \varpi]F˘=WO(k¯)[1/Ï–]. The inertia I F := Γ F ˘ I F := Γ F ˘ I_(F):=Gamma_(F^(˘))I_{F}:=\Gamma_{\breve{F}}IF:=ΓF˘ of F F FFF has a tame quotient I F t I F t I_(F)^(t)I_{F}^{t}IFt isomorphic to p Z ( 1 ) ∏ â„“ ≠ p   Z â„“ ( 1 ) prod_(â„“!=p)Z_(â„“)(1)\prod_{\ell \neq p} \mathbb{Z}_{\ell}(1)∏ℓ≠pZâ„“(1).
Theorem 1.2.2. For every choice of a topological generator τ Ï„ tau\tauÏ„ of the tame inertia I F t I F t I_(F)^(t)I_{F}^{t}IFt, there is a canonical equivalence of monoidal ∞ oo\infty∞-categories
Bez G unip : Coh ( S G ^ , Q unip ) Shv c ( H k I w k ¯ , Q ) Bez G unip  : Coh ⁡ S G ^ , Q â„“ unip  ≅ Shv c ⁡ H k I w ⊗ k ¯ , Q â„“ Bez_(G)^("unip "):Coh(S_( hat(G),Q_(â„“))^("unip "))~=Shv_(c)(Hk_(Iw)ox( bar(k)),Q_(â„“))\operatorname{Bez}_{G}^{\text {unip }}: \operatorname{Coh}\left(S_{\hat{G}, \mathbb{Q}_{\ell}}^{\text {unip }}\right) \cong \operatorname{Shv}_{c}\left(\mathrm{Hk}_{\mathrm{Iw}} \otimes \bar{k}, \mathbb{Q}_{\ell}\right)BezGunip :Coh⁡(SG^,Qâ„“unip )≅Shvc⁡(HkIw⊗k¯,Qâ„“)
In fact, Bezrukavnikov proved such equivalence when F = k ( ( ϖ ) ) F = k ( ( Ï– ) ) F=k((Ï–))F=k((\varpi))F=k((Ï–)) in [9]. Yun and the author deduced the mixed characteristic case from the equal characteristic case. It would be interesting to know whether the new techniques introduced in [25,59] can lead a direct proof of this equivalence in mixed characteristic. (See [1] for some progress in this direction.)
Remark 1.2.3. Again, for arithmetic applications, one needs to describe the action of Γ k Γ k Gamma_(k)\Gamma_{k}Γk on S h v c ( H k I w k ¯ , Λ ) S h v c H k I w ⊗ k ¯ , Λ Shv_(c)(Hk_(Iw)ox( bar(k)),Lambda)\mathbf{S h} \mathbf{v}_{c}\left(\mathrm{Hk}_{\mathrm{Iw}} \otimes \bar{k}, \Lambda\right)Shvc(HkIw⊗k¯,Λ) in terms of the dual group side. See [ 9 , 35 ] [ 9 , 35 ] [9,35][9,35][9,35] for a discussion.
We explain an important ingredient in the proof of Theorem 1.2.2 (when F = k ( ( ϖ ) ) ) F = k ( ( Ï– ) ) ) F=k((Ï–)))F=k((\varpi)))F=k((Ï–))). There is a smooth affine group scheme E E E\mathscr{E}E (called the Iwahori group scheme)

over D D DDD, analogous to H k G , D S H k G , D S Hk_(G,D^(S))\mathrm{Hk}_{G, D^{S}}HkG,DS as discussed at the end of Section 1.1 (here S = { 1 } S = { 1 } S={1}S=\{1\}S={1} ). In addition, H k G , D | D H k G , D | D H k G , D D ∗ ≅ H k G , D D ∗ Hk_(G,D)|_(D^(**))~=Hk_(G,D)|_(D^(**))\left.\left.\mathrm{Hk}_{\mathscr{G}, D}\right|_{D^{*}} \cong \mathrm{Hk}_{G, D}\right|_{D^{*}}HkG,D|D∗≅HkG,D|D∗ and H k G , D | 0 = H k I w H k G , D 0 = H k I w Hk_(G,D)|_(0)=Hk_(Iw)\left.\mathrm{Hk}_{\mathscr{G}, D}\right|_{0}=\mathrm{Hk}_{\mathrm{Iw}}HkG,D|0=HkIw, where 0 D 0 ∈ D 0in D0 \in D0∈D is the closed point. Then taking nearby cycles gives
(1.9) Z : Coh ( B G ^ Λ ) Sat 11 } Perv ( H k E , D | D k , Λ ) Ψ Perv ( H k I w k ¯ , Λ ) (1.9) Z : Coh ⁡ B G ^ Λ â—¯ → Sat 11 } Perv ⁡ H k E , D D k ∗ , Λ → Ψ Perv ⁡ H k I w ⊗ k ¯ , Λ {:(1.9)Z:Coh (B hat(G)_(Lambda))^(â—¯)rarr"Sat_(11})"Perv(Hk_(E,D)|_(D_(k)^(**)),Lambda)rarr"Psi"Perv(Hk_(Iw)ox( bar(k)),Lambda):}\begin{equation*} \mathcal{Z}: \operatorname{Coh}\left(\mathbb{B} \hat{G}_{\Lambda}\right)^{\bigcirc} \xrightarrow{\operatorname{Sat}_{11\}}} \operatorname{Perv}\left(\left.\mathrm{Hk}_{\mathscr{E}, D}\right|_{D_{k}^{*}}, \Lambda\right) \xrightarrow{\Psi} \operatorname{Perv}\left(\mathrm{Hk}_{\mathrm{Iw}} \otimes \bar{k}, \Lambda\right) \tag{1.9} \end{equation*}(1.9)Z:Coh⁡(BG^Λ)◯→Sat11}Perv⁡(HkE,D|Dk∗,Λ)→ΨPerv⁡(HkIw⊗k¯,Λ)
This is known as Gaitsgory's central functor [27,75], which can be regarded as a categorification of (1.8). We remark this construction is motivated by the Kottwitz conjecture originated from the study of mod p p ppp geometry of Shimura varieties. See Section 3.1 for some discussions.
Theorem 1.2.2 admits a generalization to the tame level. We consider the following diagram:
G ^ / G ^ B ^ / B ^ q B ^ T ^ / T ^ G ^ / G ^ ← B ^ / B ^ → q B ^ T ^ / T ^ hat(G)// hat(G)larr hat(B)// hat(B)rarr"q_( hat(B))" hat(T)// hat(T)\hat{G} / \hat{G} \leftarrow \hat{B} / \hat{B} \xrightarrow{q_{\hat{B}}} \hat{T} / \hat{T}G^/G^←B^/B^→qB^T^/T^
where the left morphism is the usual Grothendieck-Springer resolution. Let χ χ chi\chiχ be a Λ Î› Lambda\LambdaΛ-point of T ^ / T ^ T ^ / T ^ hat(T)// hat(T)\hat{T} / \hat{T}T^/T^, where Λ Î› Lambda\LambdaΛ is a finite extension of Q Q â„“ Q_(â„“)\mathbb{Q}_{\ell}Qâ„“. Let ( B ^ / B ^ ) χ = q B ^ 1 ( χ ) ( B ^ / B ^ ) χ = q B ^ − 1 ( χ ) ( hat(B)// hat(B))_(chi)=q_( hat(B))^(-1)(chi)(\hat{B} / \hat{B})_{\chi}=q_{\hat{B}}^{-1}(\chi)(B^/B^)χ=qB^−1(χ), and let
S G ^ , Λ χ := ( B ^ / B ^ ) χ × G ^ / G ^ ( B ^ / B ^ ) χ S G ^ , Λ χ := ( B ^ / B ^ ) χ × G ^ / G ^ ( B ^ / B ^ ) χ S_( hat(G),Lambda)^(chi):=( hat(B)// hat(B))_(chi)xx_( hat(G)// hat(G))( hat(B)// hat(B))_(chi)S_{\hat{G}, \Lambda}^{\chi}:=(\hat{B} / \hat{B})_{\chi} \times_{\hat{G} / \hat{G}}(\hat{B} / \hat{B})_{\chi}SG^,Λχ:=(B^/B^)χ×G^/G^(B^/B^)χ
Note that if χ = 1 χ = 1 chi=1\chi=1χ=1, this reduces to S G ^ , Λ unip S G ^ , Λ unip  S_( hat(G),Lambda)^("unip ")S_{\hat{G}, \Lambda}^{\text {unip }}SG^,Λunip . On the other hand, a (torsion) Λ Î› Lambda\LambdaΛ-point χ T ^ / T ^ χ ∈ T ^ / T ^ chi in hat(T)// hat(T)\chi \in \hat{T} / \hat{T}χ∈T^/T^ defines a one-dimensional character sheaf L χ L χ L_(chi)\mathscr{L}_{\chi}Lχ on Iw k ¯ ⊗ k ¯ ox bar(k)\otimes \bar{k}⊗k¯. Then one can define the monoidal category of bi-(Iw, L χ L χ L_(chi)\mathscr{L}_{\chi}Lχ )-equivariant constructible sheaves on L G k ¯ L G k ¯ LG_( bar(k))L G_{\bar{k}}LGk¯, denoted as Sh cons ( χ ( H k I w ) χ , Λ ) Sh cons  ⁡ χ H k I w χ , Λ Sh_("cons ")(chi(Hk_(Iw))_(chi),Lambda)\operatorname{Sh}_{\text {cons }}\left(\chi\left(\mathrm{Hk}_{\mathrm{Iw}}\right)_{\chi}, \Lambda\right)Shcons ⁡(χ(HkIw)χ,Λ). If χ = 1 χ = 1 chi=1\chi=1χ=1, so L χ L χ L_(chi)\mathscr{L}_{\chi}Lχ is the trivial character sheaf on Iw, this reduces to the affine Hecke category Shv c ( H k I w k ¯ , Λ ) Shv c ⁡ H k I w ⊗ k ¯ , Λ Shv_(c)(Hk_(Iw)ox( bar(k)),Lambda)\operatorname{Shv}_{c}\left(\mathrm{Hk}_{\mathrm{Iw}} \otimes \bar{k}, \Lambda\right)Shvc⁡(HkIw⊗k¯,Λ). The following generalization of Theorem 1.2.2 is conjectured in [9] and will be proved in a forthcoming joint work with Dhillon-Li-Yun [18].
Theorem 1.2.4. Assume that char F = char k F = char ⁡ k F=char kF=\operatorname{char} kF=char⁡k. There is a canonical monoidal equivalence
Bez G ^ χ : Coh ( S ^ G ^ , Λ χ ) Shv c ( ( H k I w ) χ , Λ ) Bez G ^ χ : Coh ⁡ S ^ G ^ , Λ χ ≅ Shv c ⁡ H k I w χ , Λ Bez_( hat(G))^(chi):Coh( hat(S)_( hat(G),Lambda)^(chi))~=Shv_(c)((Hk_(Iw))_(chi),Lambda)\operatorname{Bez}_{\hat{G}}^{\chi}: \operatorname{Coh}\left(\hat{S}_{\hat{G}, \Lambda}^{\chi}\right) \cong \operatorname{Shv}_{c}\left(\left(\mathrm{Hk}_{\mathrm{Iw}}\right)_{\chi}, \Lambda\right)BezG^χ:Coh⁡(S^G^,Λχ)≅Shvc⁡((HkIw)χ,Λ)
Remark 1.2.5. It is important to establish a version of equivalences in Theorems 1.2.2 and 1.2.4 for Z Z â„“ Z_(â„“)\mathbb{Z}_{\ell}Zâ„“-sheaves.
Remark 1.2.6. The local geometric Langlands correspondence beyond the tame ramification has not been fully understood, although certain wild ramifications have appeared in concrete problems (e.g., [31,79]). It is widely believed that the general local geometric Langlands should be formulated as 2-categorical statement, predicting the 2-category of module categories under the action of (appropriately defined) category of sheaves on L G L G LGL GLG is equivalent to the 2-category of categories over the stack of local geometric Langlands parameters. The precise formulation is beyond the scope of this survey, but, roughly speaking, it implies (and is more or less equivalent to saying) that the Hecke category for appropriately chosen "level" of L G L G LGL GLG is (Morita) equivalent to the category of coherent sheaves on some stack of the form X × Y X X × Y X Xxx_(Y)XX \times_{Y} XX×YX, where Y Y YYY is closely related to the moduli of local geometric Langlands parameters.

1.3. Global geometric Langlands correspondence

As mentioned at the beginning of the article, the (global) geometric Langlands program originated from Drinfeld's proof of Langlands conjecture for G L 2 G L 2 GL_(2)\mathrm{GL}_{2}GL2 over function fields. Early developments of this subject mostly focused on the construction of Hecke eigensheaves associated to Galois representations of a global function field F F FFF (or, equivalently, local systems on a smooth algebraic curve X X XXX ), e.g., see [ 20 , 26 , 42 ] [ 20 , 26 , 42 ] [20,26,42][20,26,42][20,26,42].
The scope of the whole program then shifted after the work [5], in which BeilinsonDrinfeld formulated a rough categorical form of the global geometric Langlands correspondence. The formulation then was made precise by Arinkin-Gaitsgory in [2], which we now recall. Let X X XXX be a smooth projective curve over F = C F = C F=CF=\mathbb{C}F=C. On the automorphic side, let D c ( B u n G ) D c B u n G D_(c)(Bun_(G))\mathbf{D}_{c}\left(\mathrm{Bun}_{G}\right)Dc(BunG) be the ∞ oo\infty∞-category of coherent D-modules on the moduli stack Bun G G _(G){ }_{G}G of principal G G GGG-bundles on X X XXX. On the Galois side, let Coh ( Loc G ^ ) Coh ⁡ Loc G ^ Coh(Loc_( hat(G)))\operatorname{Coh}\left(\operatorname{Loc}_{\hat{G}}\right)Coh⁡(LocG^) be the ∞ oo\infty∞-category of coherent sheaves on the moduli stack Loc G ^ Loc G ^ Loc_( hat(G))\operatorname{Loc}_{\hat{G}}LocG^ of de Rham G ^ G ^ hat(G)\hat{G}G^-local systems (also known as principal G ^ G ^ hat(G)\hat{G}G^-bundles with flat connection) on X X XXX.

Conjecture 1.3.1. There is a canonical equivalence of ∞ oo\infty∞-categories

L G : Coh ( Loc G ^ ) D c ( Bun G ) L G : Coh ⁡ Loc G ^ ≅ D c Bun G L_(G):Coh(Loc_( hat(G)))~=D_(c)(Bun_(G))\mathbb{L}_{G}: \operatorname{Coh}\left(\operatorname{Loc}_{\hat{G}}\right) \cong \mathbf{D}_{c}\left(\operatorname{Bun}_{G}\right)LG:Coh⁡(LocG^)≅Dc(BunG)
satisfying a list of natural compatibility conditions.
We briefly mention the most important compatibility condition, and refer to [2] for the rest. Note that both sides admit actions by a family of commuting operators labeled by x X x ∈ X x in Xx \in Xx∈X and V Coh ( B G ^ C ) V ∈ Coh ⁡ B G ^ C ∞ V in Coh (B hat(G)_(C))^(oo)V \in \operatorname{Coh}\left(\mathbb{B} \hat{G}_{\mathbb{C}}\right)^{\infty}V∈Coh⁡(BG^C)∞. Namely, for every point x X x ∈ X x in Xx \in Xx∈X, there is the evaluation map Loc G ^ B G ^ C Loc G ^ → B G ^ C Loc_( hat(G))rarrB hat(G)_(C)\operatorname{Loc}_{\hat{G}} \rightarrow \mathbb{B} \hat{G}_{\mathbb{C}}LocG^→BG^C so every V Coh ( B G ^ C ) V ∈ Coh ⁡ B G ^ C ⊙ V in Coh (B hat(G)_(C))^(o.)V \in \operatorname{Coh}\left(\mathbb{B} \hat{G}_{\mathbb{C}}\right)^{\odot}V∈Coh⁡(BG^C)⊙ gives a vector bundle V ~ x V ~ x tilde(V)_(x)\tilde{V}_{x}V~x on Loc G ^ Loc G ^ Loc_( hat(G))\operatorname{Loc}_{\hat{G}}LocG^ by pullback, which then acts on Coh ( Loc G ^ ) Coh ⁡ Loc G ^ Coh(Loc_( hat(G)))\operatorname{Coh}\left(\operatorname{Loc}_{\hat{G}}\right)Coh⁡(LocG^) by tensoring. On the other hand, there is the Hecke operator H V , x H V , x H_(V,x)H_{V, x}HV,x that acts on D c ( Bun G ) D c Bun G D_(c)(Bun_(G))\mathbf{D}_{c}\left(\operatorname{Bun}_{G}\right)Dc(BunG) by convolving the sheaf Sat { 1 } ( V ) | x Sat { 1 } ⁡ ( V ) x Sat_({1})(V)|_(x)\left.\operatorname{Sat}_{\{1\}}(V)\right|_{x}Sat{1}⁡(V)|x from the ( D D DDD-module version of) the geometric Satake (1.6). Then the equivalence L G L G L_(G)\mathbb{L}_{G}LG should intertwine the actions of these operators.
Although the conjecture remains widely open, it is known that the category of perfect complexes Perf ( Loc G ^ ) Perf ⁡ Loc G ^ Perf(Loc_( hat(G)))\operatorname{Perf}\left(\operatorname{Loc}_{\hat{G}}\right)Perf⁡(LocG^) on Loc G ^ Loc G ^ Loc_( hat(G))\operatorname{Loc}_{\hat{G}}LocG^ acts on D c ( Bun G ) D c Bun G D_(c)(Bun_(G))\mathbf{D}_{c}\left(\operatorname{Bun}_{G}\right)Dc(BunG), usually called the spectral action, in a way such that the action of V ~ x Perf ( Loc G ^ ) V ~ x ∈ Perf ⁡ Loc G ^ tilde(V)_(x)in Perf(Loc_( hat(G)))\tilde{V}_{x} \in \operatorname{Perf}\left(\operatorname{Loc}_{\hat{G}}\right)V~x∈Perf⁡(LocG^) on D c ( Bun G ) D c Bun G D_(c)(Bun_(G))\mathbf{D}_{c}\left(\operatorname{Bun}_{G}\right)Dc(BunG) is given by the Hecke operator H V , x H V , x H_(V,x)H_{V, x}HV,x.
Nowadays, Conjecture 1.3.1 sometimes is referred as the de Rham version of the global geometric Langlands conjecture, as there are some other versions of such conjectural equivalences, which we briefly mention.
First, in spirit of the nonabelian Hodge theory, there should exist a classical limit of Conjecture 1.3.1, sometimes known as the Dolbeault version of the global geometric Langlands. While the precise formulation is unknown (to the author), generically, it amounts to the duality of Hitchin fibrations for G G GGG and G ^ G ^ hat(G)\hat{G}G^ (in the sense of mirror symmetry), and was established "generically" in [15,19]. By twisting/deforming such duality in positive characteristic, one can prove a characteristic p p ppp analogue of Conjecture 1.3.1 (of course, only "generically," see [ 10 , 14 , 15 ] ) [ 10 , 14 , 15 ] ) [10,14,15])[10,14,15])[10,14,15]). Interestingly, this "generic" characteristic p p ppp equivalence can be used to give a new proof of the main result of [5] (at least for G = G L n G = G L n G=GL_(n)G=\mathrm{GL}_{n}G=GLn, see [12]).
The work [5] (and therefore the de Rham version of the global geometric Langlands) was strongly influenced by conformal field theory. On the other hand, motivated by topological field theory, Ben-Zvi and Nadler [7] proposed a Betti version of Conjecture 1.3.1, where on the automorphic side the category of D D DDD-modules on B u n G B u n G Bun_(G)\mathrm{Bun}_{G}BunG is replaced with the category of sheaves of C C C\mathbb{C}C-vector spaces on (the analytification of) Bun G G _(G){ }_{G}G and on the Galois side Loc G ^ Loc G ^ Loc_( hat(G))\operatorname{Loc}_{\hat{G}}LocG^ is replaced by the moduli of Betti G ^ G ^ hat(G)\hat{G}G^-local systems (also known as G ^ G ^ hat(G)\hat{G}G^-valued representations of fundamental group of X X XXX ).
The Riemann-Hilbert correspondence allows passing between the de Rham G ^ G ^ hat(G)\hat{G}G^-local systems and Betti G ^ G ^ hat(G)\hat{G}G^-local systems, but in a transcendental way. So Conjecture 1.3.1 and its Betti analogue are not directly related. Recently, Arinkin et al. [3] proposed yet another version of Conjecture 1.3.1, which directly relates both de Rham and Betti versions, and at the same time includes a version in terms of â„“ â„“\ellâ„“-adic sheaves. So it is more closely related to the classical Langlands correspondence over function fields, as will be discussed in Section 2.2.

2. FROM GEOMETRIC TO CLASSICAL LANGLANDS PROGRAM

In the previous section, we discussed how the ideas of categorification and geometrization led to the developments of the geometric Langlands program. On the other hand, the ideas of quantum physics allow one to reverse arrows in (1.1) by evaluating a (topological) quantum field theory at manifolds of different dimensions. Such ideas are certainly not new in geometry and topology. But, surprisingly, they also lead to a new understanding of the classical Langlands program. Indeed, it has been widely known that there is an analogy between global fields and 3-manifolds, and under such analogy Frobenius corresponds to the fundamental group of a circle. Then "compactification of field theories on a circle" leads to the categorical trace method (e.g., see [ 3 , 6 , 77 ] [ 3 , 6 , 77 ] [3,6,77][3,6,77][3,6,77] ), which plays a more and more important role in the geometric representation theory.

2.1. Categorical arithmetic local Langlands

In this subsection, let F F FFF be either a finite extension of Q p Q p Q_(p)\mathbb{Q}_{p}Qp or isomorphic to F q ( ( ϖ ) ) F q ( ( Ï– ) ) F_(q)((Ï–))\mathbb{F}_{q}((\varpi))Fq((Ï–)). The classical local Langlands correspondence seeks a classification of smooth irreducible representations of G ( F ) G ( F ) G(F)G(F)G(F) in terms of Galois data. The precise formulation, beyond the G = G L n G = G L n G=GL_(n)G=\mathrm{GL}_{n}G=GLn case (which is a theorem by [ 30 , 43 ] [ 30 , 43 ] [30,43][30,43][30,43] ), is complicated. However, the yoga that the local geometric Langlands is 2-categorical (see Remark 1.2.6) suggests that the even the classical local Langlands correspondence should and probably needs to be categorified.
The first ingredient needed to formulate the categorical arithmetic local Langlands is the following result, due independently to [ 17 , 25 , 82 ] [ 17 , 25 , 82 ] [17,25,82][17,25,82][17,25,82]. We take the formulation from [82] and refer for the notion of (strongly) continuous homomorphisms to the same reference.
Theorem 2.1.1. The prestack sending a Z Z â„“ Z_(â„“)\mathbb{Z}_{\ell}Zâ„“-algebra A A AAA to the space of (strongly) continuous homomorphisms ρ : W F c G ( A ) ρ : W F → c G ( A ) rho:W_(F)rarr^(c)G(A)\rho: W_{F} \rightarrow{ }^{c} G(A)ρ:WF→cG(A) such that d ρ = ( c y c l 1 , p r ) d ∘ ρ = c y c l − 1 , p r d@rho=(cycl^(-1),pr)d \circ \rho=\left(\mathrm{cycl}^{-1}, \mathrm{pr}\right)d∘ρ=(cycl−1,pr) is represented by a (classical) scheme Loc c Loc c â—» Loc_(c)^(â—»)\operatorname{Loc}_{c}^{\square}Loccâ—», which is a disjoint union of affine schemes that are flat, of finite type, and of locally complete intersection over Z Z â„“ Z_(â„“)\mathbb{Z}_{\ell}Zâ„“.
The conjugation action of G ^ G ^ hat(G)\hat{G}G^ on c G c G ^(c)G{ }^{c} GcG induces an action of G ^ G ^ hat(G)\hat{G}G^ on Loc c Loc c â—» Loc_(c)^(â—»)\operatorname{Loc}_{c}^{\square}Loccâ—», and we call the quotient stack Loc G = Loc c / G ^ Loc G = Loc c â—» ⁡ / G ^ Loc_(_(G))=Loc_(_(c))^(â—»)// hat(G)\operatorname{Loc}_{{ }_{G}}=\operatorname{Loc}_{{ }_{c}}^{\square} / \hat{G}LocG=Locc◻⁡/G^ the stack of local Langlands parameters, which, roughly speaking, classifies the groupoid of the above ρ ρ rho\rhoρ 's up to G ^ G ^ hat(G)\hat{G}G^-conjugacy.
In the categorical version of the local Langlands correspondence, on the Galois side it is natural to consider the ( ∞ (oo:}\left(\infty\right.(∞-)category Coh ( Loc c ) Coh ⁡ Loc c Coh(Loc_(c))\operatorname{Coh}\left(\operatorname{Loc}_{c}\right)Coh⁡(Locc) of coherent sheaves on Loc c G Loc c ⁡ G Loc_(c)_(G)\operatorname{Loc}_{c}{ }_{G}Locc⁡G. On the representation side, one might naively consider the ( ( ∞ (oo(\infty(∞-)category R e p ( G ( F ) , Λ ) R e p ( G ( F ) , Λ ) Rep(G(F),Lambda)\boldsymbol{\operatorname { R e p }}(G(F), \Lambda)R e p(G(F),Λ) of smooth representations of G ( F ) G ( F ) G(F)G(F)G(F). But in fact, this category needs to be enlarged. This can be seen from different point of view. Indeed, it is a general wisdom shared by many people that in the classical local Langlands correspondence, it is better to study representations of G G GGG together with a collection of groups that are (refined version of) its inner forms. There are various proposals of such collections. Arithmetic geometry (i.e., the study of p p ppp-adic and mod p p ppp geometry of Shimura varieties and moduli of Shtukas) and geometric representation theory (i.e., the categorical trace construction) suggest studying a category glued from the categories of representations of a collection of groups { J b ( F ) } b B ( G ) J b ( F ) b ∈ B ( G ) {J_(b)(F)}_(b in B(G))\left\{J_{b}(F)\right\}_{b \in B(G)}{Jb(F)}b∈B(G) arising from the Kottwitz set
B ( G ) = G ( F ˘ ) / , where g g if g = h 1 g σ ( h ) for some h G ( F ˘ ) B ( G ) = G ( F ˘ ) / ∼ ,  where  g ∼ g ′  if  g ′ = h − 1 g σ ( h )  for some  h ∈ G ( F ˘ ) B(G)=G(F^(˘))//∼,quad" where "g∼g^(')" if "g^(')=h^(-1)g sigma(h)" for some "h in G(F^(˘))B(G)=G(\breve{F}) / \sim, \quad \text { where } g \sim g^{\prime} \text { if } g^{\prime}=h^{-1} g \sigma(h) \text { for some } h \in G(\breve{F})B(G)=G(F˘)/∼, where g∼g′ if g′=h−1gσ(h) for some h∈G(F˘)
Here for b B ( G ) b ∈ B ( G ) b in B(G)b \in B(G)b∈B(G) (lifted to an element in G ( F ˘ ) G ( F ˘ ) G(F^(˘))G(\breve{F})G(F˘) ), the group J b J b J_(b)J_{b}Jb is an F F FFF-group defined by assigning and F F FFF-algebra the group J b ( R ) = { h G ( F ˘ F R ) h 1 b σ ( h ) = b } J b ( R ) = h ∈ G F ˘ ⊗ F R ∣ h − 1 b σ ( h ) = b J_(b)(R)={h in G((F^(˘))ox_(F)R)∣h^(-1)b sigma(h)=b}J_{b}(R)=\left\{h \in G\left(\breve{F} \otimes_{F} R\right) \mid h^{-1} b \sigma(h)=b\right\}Jb(R)={h∈G(F˘⊗FR)∣h−1bσ(h)=b}. In particular, if b = 1 b = 1 b=1b=1b=1 then J b = G J b = G J_(b)=GJ_{b}=GJb=G. In general, there is a well-defined embedding ( J b ) F ¯ G F ¯ J b F ¯ → G F ¯ (J_(b))_( bar(F))rarrG_( bar(F))\left(J_{b}\right)_{\bar{F}} \rightarrow G_{\bar{F}}(Jb)F¯→GF¯ up to conjugacy, making J b J b J_(b)J_{b}Jb a refinement of an inner form of a Levi subgroup of G G GGG (say, G G GGG is quasisplit).
There are two ways to make this idea precise. One is due to Fargues-Scholze [25], who regard B ( G ) B ( G ) B(G)B(G)B(G) as the set of points of the v v vvv-stack Bun G Bun G Bun_(G)\operatorname{Bun}_{G}BunG of G G GGG-bundles on the FarguesFontaine curve and consider the category D l i s ( Bun G , Λ ) D l i s Bun G , Λ D_(lis)(Bun_(G),Lambda)D_{\mathrm{lis}}\left(\operatorname{Bun}_{G}, \Lambda\right)Dlis(BunG,Λ) of appropriately defined étale sheaves on Bun G Bun G Bun_(G)\operatorname{Bun}_{G}BunG, which indeed glues all Rep ( J b ( F ) , Λ ) Rep ⁡ J b ( F ) , Λ Rep(J_(b)(F),Lambda)\operatorname{Rep}\left(J_{b}(F), \Lambda\right)Rep⁡(Jb(F),Λ) 's together. We mention that this approach relies on Scholze's work on â„“ â„“\ellâ„“-adic formalism of diamond and condensed mathematics.
In another approach [ 35 , 64 , 77 , 82 ] [ 35 , 64 , 77 , 82 ] [35,64,77,82][35,64,77,82][35,64,77,82], closely related to the idea of categorical trace, the set B ( G ) B ( G ) B(G)B(G)B(G) is regarded as the set of points of the (étale) quotient stack
B ( G ) := L G / Ad σ L G B ( G ) := L G / Ad σ ⁡ L G B(G):=LG//Ad_(sigma)LG\mathfrak{B}(G):=L G / \operatorname{Ad}_{\sigma} L GB(G):=LG/Adσ⁡LG
where Ad σ Ad σ Ad_(sigma)\operatorname{Ad}_{\sigma}Adσ denotes the Frobenius twisted conjugation given by
Ad σ : L G × L G L G , ( h , g ) h g σ ( h ) 1 Ad σ : L G × L G → L G , ( h , g ) ↦ h g σ ( h ) − 1 Ad_(sigma):LG xx LG rarr LG,quad(h,g)|->hg sigma(h)^(-1)\operatorname{Ad}_{\sigma}: L G \times L G \rightarrow L G, \quad(h, g) \mapsto h g \sigma(h)^{-1}Adσ:LG×LG→LG,(h,g)↦hgσ(h)−1
Then we have the category of Λ Î› Lambda\LambdaΛ-sheaves S h v ( B ( G ) k ¯ , Λ ) S h v ( B ( G ) ⊗ k ¯ , Λ ) Shv(B(G)ox bar(k),Lambda)\boldsymbol{\operatorname { S h v }}(\mathfrak{B}(G) \otimes \bar{k}, \Lambda)S h v(B(G)⊗k¯,Λ) as mentioned before. Although B ( G ) B ( G ) B(G)\mathfrak{B}(G)B(G) is a wild object in the traditional algebraic geometry, there are still a few things one can say about its geometry, and the category Shv ( B ( G ) k ¯ , Λ ) Shv ⁡ ( B ( G ) ⊗ k ¯ , Λ ) Shv(B(G)ox bar(k),Lambda)\operatorname{Shv}(\mathfrak{B}(G) \otimes \bar{k}, \Lambda)Shv⁡(B(G)⊗k¯,Λ) is quite reasonable. In addition, it is possible to define the category S h v c ( B ( G ) k ¯ , Λ ) S h v c ( B ( G ) ⊗ k ¯ , Λ ) Shv_(c)(B(G)ox bar(k),Lambda)\mathbf{S h v}_{c}(\mathfrak{B}(G) \otimes \bar{k}, \Lambda)Shvc(B(G)⊗k¯,Λ) of constructible sheaves on B ( G ) k ¯ B ( G ) ⊗ k ¯ B(G)ox bar(k)\mathfrak{B}(G) \otimes \bar{k}B(G)⊗k¯, as we now briefly explain and refer to [35] for careful discussions.
For every algebraically closed field Ω Î© Omega\OmegaΩ over k k kkk, the groupoid of Ω Î© Omega\OmegaΩ-points of B ( G ) B ( G ) B(G)\mathfrak{B}(G)B(G) is the groupoid of F F FFF-isocrystals with G G GGG-structure over Ω Î© Omega\OmegaΩ and the set of its isomorphism classes can be identified with the Kottwitz set B ( G ) B ( G ) B(G)B(G)B(G). However, B ( G ) B ( G ) B(G)\mathfrak{B}(G)B(G) is not merely a disjoint union
of its points. Rather, it admits a stratification, known as the Newton stratification, labeled by B ( G ) B ( G ) B(G)B(G)B(G). Namely, the set B ( G ) B ( G ) B(G)B(G)B(G) has a natural partial order and, roughly speaking, for each b B ( G ) b ∈ B ( G ) b in B(G)b \in B(G)b∈B(G) those Ω Î© Omega\OmegaΩ-points corresponding to b b b ′ ≤ b b^(') <= bb^{\prime} \leq bb′≤b form a closed substack i b : B ( G ) b i ≤ b : B ( G ) ≤ b ⊂ i_( <= b):B(G)_( <= b)subi_{\leq b}: \mathfrak{B}(G)_{\leq b} \subseti≤b:B(G)≤b⊂ B ( G ) k ¯ B ( G ) ⊗ k ¯ B(G)ox bar(k)\mathfrak{B}(G) \otimes \bar{k}B(G)⊗k¯ and those Ω Î© Omega\OmegaΩ-points corresponding to b b bbb form an open substack j b : B ( G ) b j b : B ( G ) b ⊂ j_(b):B(G)_(b)subj_{b}: \mathfrak{B}(G)_{b} \subsetjb:B(G)b⊂ B ( G ) b B ( G ) ≤ b B(G)_( <= b)\mathfrak{B}(G)_{\leq b}B(G)≤b. In particular, basic elements in B ( G ) B ( G ) B(G)B(G)B(G) (i.e., minimal elements with respect to the partial order ≤ <=\leq≤ ) give closed strata. We also mention that if b b bbb is basic, J b J b J_(b)J_{b}Jb is a refinement of an inner form of G G GGG, usually called an extended pure inner form of G G GGG.
In the rest of this subsection, we simply denote B ( G ) k ¯ B ( G ) ⊗ k ¯ B(G)ox bar(k)\mathfrak{B}(G) \otimes \bar{k}B(G)⊗k¯ by B ( G ) B ( G ) B(G)\mathfrak{B}(G)B(G). We write i b = i b j b : B ( G ) b B ( G ) i b = i ≤ b j b : B ( G ) b ↪ B ( G ) i_(b)=i_( <= b)j_(b):B(G)_(b)↪B(G)i_{b}=i_{\leq b} j_{b}: \mathfrak{B}(G)_{b} \hookrightarrow \mathfrak{B}(G)ib=i≤bjb:B(G)b↪B(G) for the locally closed embedding. For b b bbb, let Rep f.g. ( J b ( F ) , Λ ) Rep f.g.  ⁡ J b ( F ) , Λ Rep_("f.g. ")(J_(b)(F),Lambda)\operatorname{Rep}_{\text {f.g. }}\left(J_{b}(F), \Lambda\right)Repf.g. ⁡(Jb(F),Λ) be the full subcategory of Rep ( J b ( F ) , Λ ) Rep ⁡ J b ( F ) , Λ Rep(J_(b)(F),Lambda)\operatorname{Rep}\left(J_{b}(F), \Lambda\right)Rep⁡(Jb(F),Λ) generated (under finite colimits and retracts) by compactly induced representations
δ K , Λ := c ind K J b ( F ) ( Λ ) δ K , Λ := c − ind K J b ( F ) ⁡ ( Λ ) delta_(K,Lambda):=c-ind_(K)^(J_(b)(F))(Lambda)\delta_{K, \Lambda}:=c-\operatorname{ind}_{K}^{J_{b}(F)}(\Lambda)δK,Λ:=c−indKJb(F)⁡(Λ)
from the trivial representation of open compact subgroups K J b ( F ) K ⊂ J b ( F ) K subJ_(b)(F)K \subset J_{b}(F)K⊂Jb(F). The following theorem from [35] summarizes some properties of Shv c ( B ( G ) , Λ ) Shv c ⁡ ( B ( G ) , Λ ) Shv_(c)(B(G),Lambda)\operatorname{Shv}_{c}(\mathfrak{B}(G), \Lambda)Shvc⁡(B(G),Λ).
Theorem 2.1.2. (1) An object in Shv ( B ( G ) , Λ ) Shv ⁡ ( B ( G ) , Λ ) Shv(B(G),Lambda)\operatorname{Shv}(\mathfrak{B}(G), \Lambda)Shv⁡(B(G),Λ) is constructible if and only if its !-restriction to each B ( G ) b B ( G ) b B(G)_(b)\mathfrak{B}(G)_{b}B(G)b is constructible and is zero for almost all b's. If Λ Î› Lambda\LambdaΛ is a field of characteristic zero, Shv c ( B ( G ) , Λ ) Shv c ⁡ ( B ( G ) , Λ ) Shv_(c)(B(G),Lambda)\operatorname{Shv}_{c}(\mathfrak{B}(G), \Lambda)Shvc⁡(B(G),Λ) consist of compact objects in Shv ( B ( G ) , Λ ) Shv ⁡ ( B ( G ) , Λ ) Shv(B(G),Lambda)\operatorname{Shv}(\mathfrak{B}(G), \Lambda)Shv⁡(B(G),Λ).
(2) For every b B ( G ) b ∈ B ( G ) b in B(G)b \in B(G)b∈B(G), there is a canonical equivalence Shv c ( B ( G ) b , Λ ) Shv c ⁡ B ( G ) b , Λ ≅ Shv_(c)(B(G)_(b),Lambda)~=\operatorname{Shv}_{c}\left(\mathfrak{B}(G)_{b}, \Lambda\right) \congShvc⁡(B(G)b,Λ)≅ Rep f.g. ( J b ( F ) , Λ ) Rep f.g.  ⁡ J b ( F ) , Λ Rep_("f.g. ")(J_(b)(F),Lambda)\operatorname{Rep}_{\text {f.g. }}\left(J_{b}(F), \Lambda\right)Repf.g. ⁡(Jb(F),Λ). There are fully faithful embeddings i b , , i b , ! i b , ∗ , i b , ! i_(b,**),i_(b,!)i_{b, *}, i_{b,!}ib,∗,ib,! : Shv c ( B ( G ) b , Λ ) Shv c ( B ( G ) , Λ ) Shv c ⁡ B ( G ) b , Λ → Shv c ⁡ ( B ( G ) , Λ ) Shv_(c)(B(G)_(b),Lambda)rarrShv_(c)(B(G),Lambda)\operatorname{Shv}_{c}\left(\mathfrak{B}(G)_{b}, \Lambda\right) \rightarrow \operatorname{Shv}_{c}(\mathfrak{B}(G), \Lambda)Shvc⁡(B(G)b,Λ)→Shvc⁡(B(G),Λ) (which coincide when b b bbb is basic), inducing a semiorthogonal decomposition of Shv c ( B ( G ) , Λ ) Shv c ⁡ ( B ( G ) , Λ ) Shv_(c)(B(G),Lambda)\operatorname{Shv}_{c}(\mathfrak{B}(G), \Lambda)Shvc⁡(B(G),Λ) in terms of { S h v c ( B ( G ) b , Λ ) } b S h v c B ( G ) b , Λ b {Shv_(c)(B(G)_(b),Lambda)}_(b)\left\{\mathbf{S h v}_{c}\left(\mathfrak{B}(G)_{b}, \Lambda\right)\right\}_{b}{Shvc(B(G)b,Λ)}b.
(3) There is a self-duality functor D coh : Shv c ( B ( G ) , Λ ) S h v c ( B ( G ) , Λ ) o b D coh  : Shv c ⁡ ( B ( G ) , Λ ) ≃ S h v c ( B ( G ) , Λ ) ∨ o b − D^("coh "):Shv_(c)(B(G),Lambda)≃Shv_(c)(B(G),Lambda)^(vv)ob-\mathbb{D}^{\text {coh }}: \operatorname{Shv}_{c}(\mathfrak{B}(G), \Lambda) \simeq \mathbf{S h v}_{c}(\mathfrak{B}(G), \Lambda)^{\vee} o b-Dcoh :Shvc⁡(B(G),Λ)≃Shvc(B(G),Λ)∨ob− tained by gluing cohomological dualities (in the sense of Bernstein-Zelevinsky) on various Rep f.g. ( J b ( F ) , Λ ) Rep f.g.  ⁡ J b ( F ) , Λ Rep_("f.g. ")(J_(b)(F),Lambda)\operatorname{Rep}_{\text {f.g. }}\left(J_{b}(F), \Lambda\right)Repf.g. ⁡(Jb(F),Λ) 's.
(4) There is a natural perverse t t ttt-structure obtained by gluing (shifted) t t ttt-structures on various Rep f.g. ( J b ( F ) , Λ ) Rep f.g.  ⁡ J b ( F ) , Λ Rep_("f.g. ")(J_(b)(F),Lambda)\operatorname{Rep}_{\text {f.g. }}\left(J_{b}(F), \Lambda\right)Repf.g. ⁡(Jb(F),Λ) 's, preserved by D coh D coh  D^("coh ")\mathbb{D}^{\text {coh }}Dcoh  if Λ Î› Lambda\LambdaΛ is a field.
The following categorical form of the arithmetic local Langlands conjecture [82, SEcT. 4.6] is inspired by the global geometric Langlands conjecture as discussed in Section 1.3 .
Conjecture 2.1.3. Assume that G G GGG is quasisplit over F F FFF equipped with a pinning ( B , T , e ) ( B , T , e ) (B,T,e)(B, T, e)(B,T,e) and fix a nontrivial additive character ψ : F Z [ μ p ] × Ïˆ : F → Z â„“ μ p ∞ × psi:F rarrZ_(â„“)[mu_(p^(oo))]^(xx)\psi: F \rightarrow \mathbb{Z}_{\ell}\left[\mu_{p^{\infty}}\right]^{\times}ψ:F→Zâ„“[μp∞]×. There is a canonical equivalence of categories
Remark 2.1.4. (1) There is a closely related version of the conjecture, with Shv c ( B ( G ) , Λ ) Shv c ⁡ ( B ( G ) , Λ ) Shv_(c)(B(G),Lambda)\operatorname{Shv}_{c}(\mathfrak{B}(G), \Lambda)Shvc⁡(B(G),Λ) replaced by Shv ( B ( G ) , Λ ) Shv ⁡ ( B ( G ) , Λ ) Shv(B(G),Lambda)\operatorname{Shv}(\mathfrak{B}(G), \Lambda)Shv⁡(B(G),Λ) and with Coh ( Loc G Λ Coh ⁡ Loc G ⊗ Λ Coh(Loc_(G)ox Lambda:}\operatorname{Coh}\left(\operatorname{Loc}_{G} \otimes \Lambda\right.Coh⁡(LocG⊗Λ ) replaced by its ind-completion (with
certain support condition imposed) (see [82, SECT. 4.6]). Fargues-Scholze [25] make a conjecture parallel to this version, with the category Shv ( B ( G ) , Λ ) Shv ⁡ ( B ( G ) , Λ ) Shv(B(G),Lambda)\operatorname{Shv}(\mathfrak{B}(G), \Lambda)Shv⁡(B(G),Λ) replaced by D lis ( Bun G , Λ ) D lis  Bun G , Λ D_("lis ")(Bun_(G),Lambda)D_{\text {lis }}\left(\operatorname{Bun}_{G}, \Lambda\right)Dlis (BunG,Λ) as mentioned above.
(2) It is also explained in [82] a motivic hope to have a version of such equivalence over Q Q Q\mathbb{Q}Q.
One consequence of the conjecture is that for every b b bbb there should exist a fully faithful embedding
U J b : Rep f.g. ( J b ( F ) , Λ ) Coh ( Loc c G Λ ) U J b : Rep f.g.  ⁡ J b ( F ) , Λ → Coh ⁡ Loc c ⁡ G ⊗ Λ U_(J_(b)):Rep_("f.g. ")(J_(b)(F),Lambda)rarr Coh(Loc_(c)_(G)ox Lambda)\mathfrak{U}_{J_{b}}: \operatorname{Rep}_{\text {f.g. }}\left(J_{b}(F), \Lambda\right) \rightarrow \operatorname{Coh}\left(\operatorname{Loc}_{c}{ }_{G} \otimes \Lambda\right)UJb:Repf.g. ⁡(Jb(F),Λ)→Coh⁡(Locc⁡G⊗Λ)
obtained as the restriction of a quasinverse of L G L G L_(G)\mathbb{L}_{G}LG to i b , ! ( Rep f . g . ( J b ( F ) , Λ ) ) i b , ! Rep f . g . ⁡ J b ( F ) , Λ i_(b,!)(Rep_(f.g.)(J_(b)(F),Lambda))i_{b,!}\left(\operatorname{Rep}_{\mathrm{f} . \mathrm{g} .}\left(J_{b}(F), \Lambda\right)\right)ib,!(Repf.g.⁡(Jb(F),Λ)). The existence of such functor is closely related to the idea of local Langlands in families and has also been considered (in the case J b = G J b = G J_(b)=GJ_{b}=GJb=G is split and Λ Î› Lambda\LambdaΛ is a field of characteristic zero) in [6,32].
In particular, for every open compact subgroup K J b ( F ) K ⊂ J b ( F ) K subJ_(b)(F)K \subset J_{b}(F)K⊂Jb(F) there should exist a coherent sheaf
(2.1) V K , Λ := U J b ( δ K , Λ ) (2.1) V K , Λ := U J b δ K , Λ {:(2.1)V_(K,Lambda):=U_(J_(b))(delta_(K,Lambda)):}\begin{equation*} \mathfrak{V}_{K, \Lambda}:=\mathfrak{U}_{J_{b}}\left(\delta_{K, \Lambda}\right) \tag{2.1} \end{equation*}(2.1)VK,Λ:=UJb(δK,Λ)
on Loc G Λ Loc G ⊗ Λ Loc_(G)ox Lambda\operatorname{Loc}_{G} \otimes \LambdaLocG⊗Λ such that
The algebra H K , Λ H K , Λ H_(K,Lambda)H_{K, \Lambda}HK,Λ is sometimes called the derived Hecke algebra as it might not concentrate on cohomological degree zero (when Λ = Z Λ = Z â„“ Lambda=Z_(â„“)\Lambda=\mathbb{Z}_{\ell}Λ=Zâ„“ or F F â„“ F_(â„“)\mathbb{F}_{\ell}Fâ„“ ). See [82, SECTS. 4.3-4.5] for conjectural descriptions of A K , Λ A K , Λ A_(K,Lambda)\mathfrak{A}_{K, \Lambda}AK,Λ in various cases.
As in the global geometric Langlands conjecture, the equivalence from Conjecture 2.1.3 should satisfy a set of compatibility conditions. For example, it should be compatible with parabolic inductions on both sides, and should be compatible with the duality D coh D coh  D^("coh ")\mathbb{D}^{\text {coh }}Dcoh  on Shv c ( B ( G ) , Λ ) Shv c ⁡ ( B ( G ) , Λ ) Shv_(c)(B(G),Lambda)\operatorname{Shv}_{c}(\mathfrak{B}(G), \Lambda)Shvc⁡(B(G),Λ) and the (modified) Grothendieck-Serre duality of Coh ( Loc c Λ ) Coh ⁡ Loc c ⊗ Λ Coh(Loc_(c)ox Lambda)\operatorname{Coh}\left(\operatorname{Loc}_{c} \otimes \Lambda\right)Coh⁡(Locc⊗Λ). We refer to [ 35 , 82 ] [ 35 , 82 ] [35,82][35,82][35,82] for more details.
On the other hand, Conjecture 2.1.3 predicts an action of the category Perf ( Loc G Λ ) Perf ⁡ Loc G ⊗ Λ Perf(Loc_(G)ox Lambda)\operatorname{Perf}\left(\operatorname{Loc}_{G} \otimes \Lambda\right)Perf⁡(LocG⊗Λ) of perfect complexes on Loc G Λ Loc G ⊗ Λ Loc_(G)ox Lambda\operatorname{Loc}_{G} \otimes \LambdaLocG⊗Λ on Shv c ( B ( G ) , Λ ) Shv c ⁡ ( B ( G ) , Λ ) Shv_(c)(B(G),Lambda)\operatorname{Shv}_{c}(\mathfrak{B}(G), \Lambda)Shvc⁡(B(G),Λ), analogous to the spectral action as mentioned in Section 1.3. One of the main results of [25] is the construction of such action in their setting. Currently the existence of such a spectral action on Shv c ( B ( G ) , Λ ) Shv c ⁡ ( B ( G ) , Λ ) Shv_(c)(B(G),Lambda)\operatorname{Shv}_{c}(\mathfrak{B}(G), \Lambda)Shvc⁡(B(G),Λ) is not known. But there are convincing evidences that Conjecture 2.1.3 should still be true.
We assume that G G GGG extends to a reductive group over O O O\mathcal{O}O as before. Then there are closed substacks
usually called the stack of unramified parameters (resp. unipotent parameters), classifying those ρ ρ rho\rhoρ such that ρ ( I F ) ρ I F rho(I_(F))\rho\left(I_{F}\right)ρ(IF) is trivial (resp. ρ ( I F ) ρ I F rho(I_(F))\rho\left(I_{F}\right)ρ(IF) is unipotent). For Λ = Q , Loc c unip Q Λ = Q â„“ , Loc c unip  ⊗ Q â„“ Lambda=Q_(â„“),Loc_(c)^("unip ")oxQ_(â„“)\Lambda=\mathbb{Q}_{\ell}, \operatorname{Loc}_{c}^{\text {unip }} \otimes \mathbb{Q}_{\ell}Λ=Qâ„“,Loccunip ⊗Qâ„“ is a connected component of Loc G G Q Loc G ⁡ G ⊗ Q â„“ Loc^(G)_(G)oxQ_(â„“)\operatorname{Loc}^{G}{ }_{G} \otimes \mathbb{Q}_{\ell}LocG⁡G⊗Qâ„“.
On the other hand, there is the unipotent subcategory Shv c unip ( B ( G ) , Q ) Shv c unip  ⁡ B ( G ) , Q ℓ ⊂ Shv_(c)^("unip ")(B(G),Q_(ℓ))sub\operatorname{Shv}_{c}^{\text {unip }}\left(\mathfrak{B}(G), \mathbb{Q}_{\ell}\right) \subsetShvcunip ⁡(B(G),Qℓ)⊂ Sh c ( B ( G ) , Q ) Sh c ⁡ B ( G ) , Q ℓ Sh_(c)(B(G),Q_(ℓ))\operatorname{Sh}_{c}\left(\mathfrak{B}(G), \mathbb{Q}_{\ell}\right)Shc⁡(B(G),Qℓ), which roughly speaking is the glue of categories Rep f.g. unip ( J b ( F ) , Q ) Rep f.g.  unip  ⁡ J b ( F ) , Q ℓ Rep_("f.g. ")^("unip ")(J_(b)(F),Q_(ℓ))\operatorname{Rep}_{\text {f.g. }}^{\text {unip }}\left(J_{b}(F), \mathbb{Q}_{\ell}\right)Repf.g. unip ⁡(Jb(F),Qℓ)
of unipotent representations of J b ( F ) J b ( F ) J_(b)(F)J_{b}(F)Jb(F) (introduced in [52]) for all b B ( G ) b ∈ B ( G ) b in B(G)b \in B(G)b∈B(G). We have the following theorem from [35], deduced from Theorem 1.2.2 by taking the Frobenius-twisted categorical trace.
Theorem 2.1.5. For a reductive group G G GGG over O O O\mathcal{O}O with a fixed pinning ( B , T , e ) ( B , T , e ) (B,T,e)(B, T, e)(B,T,e), there is a canonical equivalence
L G unip : Coh ( Loc c unip Q ) Sh c unip ( B ( G ) , Q ) L G unip  : Coh ⁡ Loc c unip  ⊗ Q ℓ ≅ Sh c unip  ⁡ B ( G ) , Q ℓ L_(G)^("unip "):Coh(Loc_(c)^("unip ")oxQ_(ℓ))~=Sh_(c)^("unip ")(B(G),Q_(ℓ))\mathbb{L}_{G}^{\text {unip }}: \operatorname{Coh}\left(\operatorname{Loc}_{c}^{\text {unip }} \otimes \mathbb{Q}_{\ell}\right) \cong \operatorname{Sh}_{c}^{\text {unip }}\left(\mathfrak{B}(G), \mathbb{Q}_{\ell}\right)LGunip :Coh⁡(Loccunip ⊗Qℓ)≅Shcunip ⁡(B(G),Qℓ)
For arithmetic applications, it is important to match specific objects under the equivalence. We give a few examples and refer to [35] for many more of such matchings (see also [82, SECTS. 4.3-4.5]).
Example 2.1.6. The equivalence L G unip L G unip  L_(G)^("unip ")\mathbb{L}_{G}^{\text {unip }}LGunip  gives the the conjectural coherent sheaf in (2.1) for all parahoric subgroups K G ( F ) K ⊂ G ( F ) K sub G(F)K \subset G(F)K⊂G(F) (in the sense of Bruhat-Tits) such that (2.2) holds. For example, we have V G ( O ) , Q O Loc C G u r Q V G ( O ) , Q ℓ ≅ O Loc  C G u r ⊗ Q ℓ V_(G(O),Q_(ℓ))~=O_("Loc "_(C_(G))^(ur))oxQ_(ℓ)\mathfrak{V}_{G(\mathcal{O}), \mathbb{Q}_{\ell}} \cong \mathcal{O}_{\text {Loc }_{C_{G}}^{\mathrm{ur}}} \otimes \mathbb{Q}_{\ell}VG(O),Qℓ≅OLoc CGur⊗Qℓ, which gives
(2.3) ( R End Coh ( Loc G ) O Loc C G u r ) o p Q ( R End δ G ( O ) , Q ) o p = H G ( O ) , Q H G ( O ) c l Q (2.3) R End Coh ⁡ Loc G ⁡ O Loc C G u r ) o p ⊗ Q â„“ ≅ R End ⁡ δ G ( O ) , Q â„“ o p = H G ( O ) , Q â„“ ≅ H G ( O ) c l ⊗ Q â„“ {:(2.3)(REnd_(Coh(Loc_(G)))O_(Loc_(CG)^(ur))^()^(op))oxQ_(â„“)~=(R End delta_(G(O),Q_(â„“)))^(op)=H_(G(O),Q_(â„“))~=H_(G(O))^(cl)oxQ_(â„“):}:}\begin{equation*} \left(R \operatorname{End}_{\operatorname{Coh}\left(\operatorname{Loc}_{G}\right)} \mathcal{O}_{\operatorname{Loc}_{C G}^{\mathrm{ur}}}^{)^{\mathrm{op}}} \otimes \mathbb{Q}_{\ell} \cong\left(R \operatorname{End} \delta_{G(\mathcal{O}), \mathbb{Q}_{\ell}}\right)^{\mathrm{op}}=H_{G(\mathcal{O}), \mathbb{Q}_{\ell}} \cong H_{G(\mathcal{O})}^{\mathrm{cl}} \otimes \mathbb{Q}_{\ell}\right. \tag{2.3} \end{equation*}(2.3)(REndCoh⁡(LocG)⁡OLocCGur)op⊗Qℓ≅(REnd⁡δG(O),Qâ„“)op=HG(O),Qℓ≅HG(O)cl⊗Qâ„“
As Loc c G u r ( c G | d = ( q , σ ) ) / G ^ Loc c G u r ≅ c G d = ( q , σ ) / G ^ Loc_(cG)^(ur)~=(^(c)G|_(d=(q,sigma)))// hat(G)\operatorname{Loc}_{c G}^{\mathrm{ur}} \cong\left(\left.{ }^{c} G\right|_{d=(q, \sigma)}\right) / \hat{G}LoccGur≅(cG|d=(q,σ))/G^, taking the 0 th cohomology recovers the Satake isomorphism (1.3). In addition, it implies that the left-hand side has no higher cohomology, which is not obvious. We mention that it is conjectured in [82, SECT. 4.3] that V G ( O ) , Z O Loc c G V G ( O ) , Z â„“ ≅ O Loc  c G V_(G(O),Z_(â„“))~=O_("Loc "_(c_(G)))\mathfrak{V}_{G(\mathcal{O}), \mathbb{Z}_{\ell}} \cong \mathcal{O}_{\text {Loc }_{c_{G}}}VG(O),Zℓ≅OLoc cG so the first isomorphism in (2.3) should hold over Z Z â„“ Z_(â„“)\mathbb{Z}_{\ell}Zâ„“, known as the (conjectural) derived Satake isomorphism. (But H G ( O ) , Z H G ( O ) c l Z H G ( O ) , Z â„“ ≠ H G ( O ) c l ⊗ Z â„“ H_(G(O),Z_(â„“))!=H_(G(O))^(cl)oxZ_(â„“)H_{G(\mathcal{O}), \mathbb{Z}_{\ell}} \neq H_{G(\mathcal{O})}^{\mathrm{cl}} \otimes \mathbb{Z}_{\ell}HG(O),Zℓ≠HG(O)cl⊗Zâ„“ in general.)
There is also a pure Galois side description of A I , Q A I , Q â„“ A_(I,Q_(â„“))\mathfrak{A}_{I, \mathbb{Q}_{\ell}}AI,Qâ„“, known as the unipotent coherent Springer sheaf as defined in [6,82] (see also [32]).
Example 2.1.7. By construction, there is a natural morphism of stacks Loc G B G ^ Loc G → B G ^ Loc_(G)rarrB hat(G)\operatorname{Loc}_{G} \rightarrow \mathbb{B} \hat{G}LocG→BG^ over Z Z â„“ Z_(â„“)\mathbb{Z}_{\ell}Zâ„“. For a representation of G ^ G ^ hat(G)\hat{G}G^ on a finite projective Λ Î› Lambda\LambdaΛ-module, regarded as a vector bundle on B G ^ Λ B G ^ Λ B hat(G)_(Lambda)\mathbb{B} \hat{G}_{\Lambda}BG^Λ, let V ~ V ~ tilde(V)\tilde{V}V~ be its pullback to Loc G Λ Loc G ⊗ Λ Loc_(G)ox Lambda\operatorname{Loc}_{G} \otimes \LambdaLocG⊗Λ, and let V ~ ? Perf ( Loc c G Λ ) V ~ ? ∈ Perf ⁡ Loc c G ⊗ Λ tilde(V)^(?)in Perf(Loc_(c_(G))ox Lambda)\tilde{V}{ }^{?} \in \operatorname{Perf}\left(\operatorname{Loc}_{c_{G}} \otimes \Lambda\right)V~?∈Perf⁡(LoccG⊗Λ) be its

We have
L G u n i p ( V ~ u r ) N t ! r ! Sat ( V ) =: Φ V L G u n i p V ~ u r ≅ N t ! r ! Sat ⁡ ( V ) =: Φ V L_(G)^(unip)( tilde(V)^(ur))~=Nt_(!)r!Sat(V)=:Phi_(V)\mathbb{L}_{G}^{\mathrm{unip}}\left(\tilde{V}^{\mathrm{ur}}\right) \cong \mathrm{Nt}_{!} r!\operatorname{Sat}(V)=: \Phi_{V}LGunip(V~ur)≅Nt!r!Sat⁡(V)=:ΦV
where r r rrr and N t N t Nt\mathrm{Nt}Nt are maps in the following correspondence:
H k G = L + G L G / L + G r L G / Ad σ L + G N t L G / Ad σ L G = B ( G ) . H k G = L + G ∖ L G / L + G ← r L G / Ad σ ⁡ L + G → N t L G / Ad σ ⁡ L G = B ( G ) . Hk_(G)=L^(+)G\\LG//L^(+)Glarr^(r)LG//Ad_(sigma)L^(+)Grarr"Nt"LG//Ad_(sigma)LG=B(G).\mathrm{Hk}_{G}=L^{+} G \backslash L G / L^{+} G \stackrel{r}{\leftarrow} L G / \operatorname{Ad}_{\sigma} L^{+} G \xrightarrow{\mathrm{Nt}} L G / \operatorname{Ad}_{\sigma} L G=\mathfrak{B}(G) .HkG=L+G∖LG/L+G←rLG/Adσ⁡L+G→NtLG/Adσ⁡LG=B(G).
In particular, for two representations V V VVV and W W WWW of G ^ G ^ hat(G)\hat{G}G^, there is a morphism
(2.4) R Hom L o c C G u r Q ( V ~ u r , W ~ u r ) R Hom Shv c ( B ( G ) , Q ) ( S V , ς W ) (2.4) R Hom L o c C G u r ⊗ Q â„“ V ~ u r , W ~ u r → R Hom Shv c ⁡ B ( G ) , Q â„“ S V , Ï‚ W {:(2.4){:RHom_(Loc_(CG)^(ur))oxQ_(â„“)( tilde(V)^(ur), tilde(W)^(ur))rarr RHom_(Shv_(c)(B(G),Q_(â„“):}))(S_(V),Ï‚_(W)):}\begin{equation*} \left.R \operatorname{Hom}_{\mathrm{Loc}_{C G}^{\mathrm{ur}}} \otimes \mathbb{Q}_{\ell}\left(\tilde{V}^{\mathrm{ur}}, \tilde{W}^{\mathrm{ur}}\right) \rightarrow R \operatorname{Hom}_{\operatorname{Shv}_{c}\left(\mathfrak{B}(G), \mathbb{Q}_{\ell}\right.}\right)\left(S_{V}, \varsigma_{W}\right) \tag{2.4} \end{equation*}(2.4)RHomLocCGur⊗Qâ„“(V~ur,W~ur)→RHomShvc⁡(B(G),Qâ„“)(SV,Ï‚W)
compatible with compositions. Such map was first constructed in [64,77] and (the version for underived Hom spaces) was then extended to Z Z â„“ Z_(â„“)\mathbb{Z}_{\ell}Zâ„“-coefficient in [70]. It has significant arithmetic applications, as will be explained in Section 3.
Remark 2.1.8. It is likely that Theorem 2.1 .5 can be extended to the tame level by taking the Frobenius-twisted categorical trace of the equivalence from Theorem 1.2.4. On the other hand, as mentioned in Remark 1.2.5, it is important to extend these equivalences to Z Z ℓ − Z_(ℓ^(-))\mathbb{Z}_{\ell^{-}}Zℓ− coefficient.

2.2. Global arithmetic Langlands for function fields

Next we turn to global aspects of the arithmetic Langlands correspondence. As mentioned at the beginning, its classical formulation, very roughly speaking, predicts a natural correspondence between the set of (irreducible) Galois representations and the set of (cuspidal) automorphic representations. As in the local case, beyond the G L n G L n GL_(n)\mathrm{GL}_{n}GLn case (which is a theorem by [38]), such a formulation is not easy to be made precise. On the other hand, the global geometric Langlands conjecture from Section 1.3 and philosophy of decategorification/trace suggest that the global arithmetic Langlands can and probably should be formulated as an isomorphism between two vector spaces, arising from the Galois and the automorphic side, respectively. In this subsection, we formulate such a conjecture in the global function field case.
Let F = F q ( X ) F = F q ( X ) F=F_(q)(X)F=\mathbb{F}_{q}(X)F=Fq(X) be the function field of a geometrically connected smooth projective curve X X XXX over F q F q F_(q)\mathbb{F}_{q}Fq. We write η = Spec F η = Spec ⁡ F eta=Spec F\eta=\operatorname{Spec} Fη=Spec⁡F for the generic point of X X XXX and η ¯ η ¯ bar(eta)\bar{\eta}η¯ for a geometric point over η η eta\etaη. Let | X | | X | |X||X||X| denote the set of closed points of X X XXX. For v | X | v ∈ | X | v in|X|v \in|X|v∈|X|, let O v O v O_(v)\mathcal{O}_{v}Ov denote the complete local ring of X X XXX at v v vvv and F v F v F_(v)F_{v}Fv its fractional field. Let O F = v | X | O v O F = ∏ v ∈ | X |   O v O_(F)=prod_(v in|X|)O_(v)\mathbb{O}_{F}=\prod_{v \in|X|} \mathcal{O}_{v}OF=∏v∈|X|Ov be the integral adèles, and A F = v | X | F v A F = ∏ v ∈ | X | ′   F v A_(F)=prod_(v in|X|)^(')F_(v)\mathbb{A}_{F}=\prod_{v \in|X|}^{\prime} F_{v}AF=∏v∈|X|′Fv the ring of adèles. For a finite nonempty set of places Q Q QQQ, let W F , Q W F , Q W_(F,Q)W_{F, Q}WF,Q denote the Weil group of F F FFF, unramified outside Q Q QQQ.
Let G G GGG be a connected reductive group over F F FFF. Similarly to the local situation, the first step to formulate our global conjecture is the following theorem from [82].
Theorem 2.2.1. Assume that 2 p â„“ ∤ 2 p ℓ∤2p\ell \nmid 2 pℓ∤2p. The prestack sending a Z a Z â„“ aZ_(â„“)a \mathbb{Z}_{\ell}aZâ„“-algebra A A AAA to the space of (strongly) continuous homomorphisms ρ : W F , Q c G ( A ) ρ : W F , Q → c G ( A ) rho:W_(F,Q)rarr^(c)G(A)\rho: W_{F, Q} \rightarrow{ }^{c} G(A)ρ:WF,Q→cG(A) such that d ρ = ( c y c l 1 , p r ) d ∘ ρ = c y c l − 1 , p r d@rho=(cycl^(-1),pr)d \circ \rho=\left(\mathrm{cycl}^{-1}, \mathrm{pr}\right)d∘ρ=(cycl−1,pr) is represented by a derived scheme Loc c Loc c â—» Loc_(c)^(â—»)\operatorname{Loc}_{c}^{\square}Loccâ—», , which is a disjoint union of derived affine schemes that are flat and of finite type over Z Z â„“ Z_(â„“)\mathbb{Z}_{\ell}Zâ„“. If Q , Loc c G , Q Q ≠ ∅ , Loc c G , Q â—» Q!=O/,Loc_(cG,Q)^(â—»)Q \neq \emptyset, \operatorname{Loc}_{c G, Q}^{\square}Q≠∅,LoccG,Qâ—» is quasismooth.
We then define the stack of global Langlands parameters as Loc G , Q = Loc G , Q / G ^ Loc G , Q = Loc G , Q ⁡ / G ^ Loc_(G,Q)=Loc_(_(G),Q)// hat(G)\operatorname{Loc}_{G, Q}=\operatorname{Loc}_{{ }_{G}, Q} / \hat{G}LocG,Q=LocG,Q⁡/G^. Similar to the local case (see Example 2.1.7), for a representation of G ^ Λ G ^ Λ hat(G)_(Lambda)\hat{G}_{\Lambda}G^Λ on a finite projective Λ Î› Lambda\LambdaΛ-module, regarded as a vector bundle on B G ^ Λ B G ^ Λ B hat(G)_(Lambda)\mathbb{B} \hat{G}_{\Lambda}BG^Λ, let V ~ V ~ tilde(V)\tilde{V}V~ be its pullback to Loc G , Q Λ Loc G , Q ⊗ Λ Loc_(G,Q)ox Lambda\operatorname{Loc}_{G, Q} \otimes \LambdaLocG,Q⊗Λ. If V V VVV is the restriction of a representation of ( c G ) S c G S (^(c)G)^(S)\left({ }^{c} G\right)^{S}(cG)S along the diagonal embedding G ^ ( c G ) S G ^ → c G S hat(G)rarr(^(c)G)^(S)\hat{G} \rightarrow\left({ }^{c} G\right)^{S}G^→(cG)S, then there is a natural (strongly) continuous W F , Q S W F , Q S W_(F,Q)^(S)W_{F, Q}^{S}WF,QS-action on V ~ V ~ tilde(V)\tilde{V}V~ (see [82, SEcT. 2.4]). For a place v v vvv of F F FFF, let Loc c G , v Loc c G , v Loc_(_(c)G,v)\operatorname{Loc}_{{ }_{c} G, v}LoccG,v denote the stack of local Langlands parameters for G F v G F v G_(F_(v))G_{F_{v}}GFv. Let
res : Loc G , Q v Q Loc G , v  res  : Loc G , Q → ∏ v ∈ Q   Loc G , v " res ":Loc_(G,Q)rarrprod_(v in Q)Loc_(G,v)\text { res }: \operatorname{Loc}_{G, Q} \rightarrow \prod_{v \in Q} \operatorname{Loc}_{G, v} res :LocG,Q→∏v∈QLocG,v
denote the map by restricting global parameters to local parameters (induced by the map W F v W F , Q ) W F v → W F , Q {:W_(F_(v))rarrW_(F,Q))\left.W_{F_{v}} \rightarrow W_{F, Q}\right)WFv→WF,Q). Later on, we will consider the !-pullback of coherent sheaves on v Q Loc c , v ∏ v ∈ Q   Loc c , v prod_(v in Q)Loc_(c,v)\prod_{v \in Q} \operatorname{Loc}_{c, v}∏v∈QLocc,v along this map.
Remark 2.2.2. (1) In fact, when Q = Q = ∅ Q=O/Q=\emptysetQ=∅, the definition of Loc G , Q Loc G , Q Loc_(G,Q)\operatorname{Loc}_{G, Q}LocG,Q needs to be slightly modified.
(2) Unlike the local situation, Loc G , Q Loc G , Q Loc_(G),Q\operatorname{Loc}_{G}, QLocG,Q has nontrivial derived structure in general (see [82, REMARK 3.4.5]). Let c l Loc G , Q Q c l Loc G , Q ⁡ Q ^(cl)Loc^(G,Q)^(Q){ }^{c l} \operatorname{Loc}^{G, Q}{ }^{Q}clLocG,Q⁡Q denote the underlying classical stack.
(3) A different definition of Loc G , Q Q Loc G , Q ⊗ Q ℓ Loc_(G,Q)oxQ_(ℓ)\operatorname{Loc}_{G, Q} \otimes \mathbb{Q}_{\ell}LocG,Q⊗Qℓ is given by [3].
Next we move to the automorphic side. For simplicity, we assume that G G GGG is split over F q F q F_(q)\mathbb{F}_{q}Fq in this subsection. Fix a level, i.e., an open compact subgroup K G ( O F ) K ⊂ G O F K sub G(O_(F))K \subset G\left(\mathbb{O}_{F}\right)K⊂G(OF). Let Q Q QQQ be the set of places consisting of those v v vvv such that K v G ( O v ) K v ≠ G O v K_(v)!=G(O_(v))K_{v} \neq G\left(\mathcal{O}_{v}\right)Kv≠G(Ov). For a finite set S S SSS, let Sht K ( G ) ( X Q ) S K ( G ) ( X − Q ) S _(K)(G)_((X-Q)^(S))_{K}(G)_{(X-Q)^{S}}K(G)(X−Q)S denote the ind-Deligne-Mumford stack over ( X Q ) S ( X − Q ) S (X-Q)^(S)(X-Q)^{S}(X−Q)S of the moduli of G G GGG-shtukas on X X XXX with S S SSS-legs in X Q X − Q X-QX-QX−Q and K K KKK-level structure. (For example, see [39] for basic constructions and properties of this moduli space.) Its base change along the diagonal map η ¯ ( X Q ) Δ ( X Q ) S η ¯ → ( X − Q ) → Δ ( X − Q ) S bar(eta)rarr(X-Q)rarr"Delta"(X-Q)^(S)\bar{\eta} \rightarrow(X-Q) \xrightarrow{\Delta}(X-Q)^{S}η¯→(X−Q)→Δ(X−Q)S is denoted by Sht K ( G ) Δ ( η ¯ ) Sht K ⁡ ( G ) Δ ( η ¯ ) Sht_(K)(G)_(Delta( bar(eta)))\operatorname{Sht}_{K}(G)_{\Delta(\bar{\eta})}ShtK⁡(G)Δ(η¯). For every representation V V VVV of ( c G ) S c G S (^(c)G)^(S)\left({ }^{c} G\right)^{S}(cG)S on a finite projective Λ Î› Lambda\LambdaΛ-module, the geometric Satake (1.6) (with D D DDD replaced by X Q X − Q X-QX-QX−Q and with Λ = Z Λ = Z â„“ Lambda=Z_(â„“)\Lambda=\mathbb{Z}_{\ell}Λ=Zâ„“ allowed) provides a perverse sheaf Sat S ( V ) Sat S ⁡ ( V ) Sat_(S)(V)\operatorname{Sat}_{S}(V)SatS⁡(V) on Sht K ( G ) ( X Q ) S Sht K ⁡ ( G ) ( X − Q ) S Sht_(K)(G)_((X-Q)^(S))\operatorname{Sht}_{K}(G)_{(X-Q)^{S}}ShtK⁡(G)(X−Q)S. Let C c ( Sht K ( G ) Δ ( η ¯ ) C c Sht K ⁡ ( G ) Δ ( η ¯ ) C_(c)(Sht_(K)(G)_(Delta( bar(eta))):}C_{c}\left(\operatorname{Sht}_{K}(G)_{\Delta(\bar{\eta})}\right.Cc(ShtK⁡(G)Δ(η¯), Sat S S ( V ) ) S S ( V ) {:S_(S)(V))\left.S_{S}(V)\right)SS(V)) denote the (cochain complex of the) total compactly supported cohomology of Sht K ( G ) Δ ( η ¯ ) Sht K ⁡ ( G ) Δ ( η ¯ ) Sht_(K)(G)_(Delta( bar(eta)))\operatorname{Sht}_{K}(G)_{\Delta(\bar{\eta})}ShtK⁡(G)Δ(η¯) with coefficient in Sat S ( V ) Sat S ⁡ ( V ) Sat_(S)(V)\operatorname{Sat}_{S}(V)SatS⁡(V). It admits a (strongly) continuous action of W F , Q S W F , Q S W_(F,Q)^(S)W_{F, Q}^{S}WF,QS (see [34] for the construction of such action at the derived level, based on [ 67 , 68 ] [ 67 , 68 ] [67,68][67,68][67,68] ), as well as an action of the corresponding global (derived) Hecke algebra (with coefficients in Λ Î› Lambda\LambdaΛ )
(2.5) H K , Λ = ( R End ( c ind K G ( A F ) ( Λ ) ) ) o p (2.5) H K , Λ = R End ⁡ c − ind K G A F ⁡ ( Λ ) o p {:(2.5)H_(K,Lambda)=(R End(c-ind_(K)^(G(A_(F)))(Lambda)))^(op):}\begin{equation*} H_{K, \Lambda}=\left(R \operatorname{End}\left(c-\operatorname{ind}_{K}^{G\left(\mathbb{A}_{F}\right)}(\Lambda)\right)\right)^{\mathrm{op}} \tag{2.5} \end{equation*}(2.5)HK,Λ=(REnd⁡(c−indKG(AF)⁡(Λ)))op
For example, if V = 1 V = 1 V=1V=\mathbf{1}V=1 is the trivial representation, then (under our assumption that G G GGG is split)
C c ( Sht K ( G ) Δ ( η ¯ ) , Sat { 1 } ( 1 ) ) = C c ( G ( F ) G ( A ) / K , Λ ) C c Sht K ⁡ ( G ) Δ ( η ¯ ) , Sat { 1 } ⁡ ( 1 ) = C c ( G ( F ) ∖ G ( A ) / K , Λ ) C_(c)(Sht_(K)(G)_(Delta( bar(eta))),Sat_({1})(1))=C_(c)(G(F)\\G(A)//K,Lambda)C_{c}\left(\operatorname{Sht}_{K}(G)_{\Delta(\bar{\eta})}, \operatorname{Sat}_{\{1\}}(\mathbf{1})\right)=C_{c}(G(F) \backslash G(\mathbb{A}) / K, \Lambda)Cc(ShtK⁡(G)Δ(η¯),Sat{1}⁡(1))=Cc(G(F)∖G(A)/K,Λ)
Here G ( F ) G ( A ) / K G ( F ) ∖ G ( A ) / K G(F)\\G(A)//KG(F) \backslash G(\mathbb{A}) / KG(F)∖G(A)/K is regarded as a discrete DM stack over η ¯ η ¯ bar(eta)\bar{\eta}η¯, and C c ( G ( F ) G ( A ) / K , Λ ) C c ( G ( F ) ∖ G ( A ) / K , Λ ) C_(c)(G(F)\\G(A)//K,Lambda)C_{c}(G(F) \backslash G(\mathbb{A}) / K, \Lambda)Cc(G(F)∖G(A)/K,Λ) denotes its compactly supported cohomology. When Λ = Q Λ = Q â„“ Lambda=Q_(â„“)\Lambda=\mathbb{Q}_{\ell}Λ=Qâ„“, this is the space of compactly supported functions on G ( F ) G ( A ) / K G ( F ) ∖ G ( A ) / K G(F)\\G(A)//KG(F) \backslash G(\mathbb{A}) / KG(F)∖G(A)/K.
We will fix a pinning ( B , T , e ) ( B , T , e ) (B,T,e)(B, T, e)(B,T,e) of G G GGG and a nondegenerate character ψ : F A ψ : F ∖ A → psi:F\\Ararr\psi: F \backslash \mathbb{A} \rightarrowψ:F∖A→ Z [ μ p ] × Z â„“ μ p × Z_(â„“)[mu_(p)]^(xx)\mathbb{Z}_{\ell}\left[\mu_{p}\right]^{\times}Zâ„“[μp]×, which gives the conjectural equivalence L v L v L_(v)\mathbb{L}_{v}Lv as in Conjecture 2.1.3 for every v Q v ∈ Q v in Qv \in Qv∈Q. In particular, corresponding to K v G ( F v ) K v ⊂ G F v K_(v)sub G(F_(v))K_{v} \subset G\left(F_{v}\right)Kv⊂G(Fv) there is a conjectural coherent sheaf N K v N K v N_(K_(v))\mathfrak{N}_{K_{v}}NKv (see (2.1)) on Loc G , v Loc G , v Loc_(G,v)\operatorname{Loc}_{G, v}LocG,v.
Conjecture 2.2.3. There is a natural ( W F , Q S × H K , Λ ) W F , Q S × H K , Λ (W_(F,Q)^(S)xxH_(K,Lambda))\left(W_{F, Q}^{S} \times H_{K, \Lambda}\right)(WF,QS×HK,Λ)-equivariant isomorphism
We refer to [82, SECT. 4.7] for more general form of the conjecture (where "generalized level structures" are allowed) and examples of such conjecture in various special cases. This conjecture could be regarded a precise form of the global Langlands correspondence for function fields. Namely, it gives a precise recipe to match Galois representations and automorphic representations. (For example, V. Lafforgue's excursion operators are encoded in such isomorphism, see below.) Moreover, such an isomorphism fits in the Arthur-Kottwitz multiplicity formula and at the same time extends such a formula to the integral level and therefore relates to automorphic lifting theories.
The most appealing evidence of this conjecture is the following theorem [40,82], as suggested (at the heuristic level) by Drinfeld as an interpretation of Lafforgue's construction.

with an action of H K , Q H K , Q â„“ H_(K,Q_(â„“))H_{K, \mathbb{Q}_{\ell}}HK,Qâ„“, such that for every finite dimensional Q Q â„“ Q_(â„“)\mathbb{Q}_{\ell}Qâ„“-representation V V VVV of ( c G ) S c G S (^(c)G)^(S)\left({ }^{c} G\right)^{S}(cG)S, there is a natural ( W F , Q S × H K , Q ) W F , Q S × H K , Q â„“ (W_(F,Q)^(S)xxH_(K,Q_(â„“)))\left(W_{F, Q}^{S} \times H_{K, \mathbb{Q}_{\ell}}\right)(WF,QS×HK,Qâ„“)-equivariant isomorphism
We mention that this theorem actually was proved for any G G GGG in [40,82]. In addition, when K K KKK is everywhere hyperspecial, (2.2.4) holds at the derived level by [3].
The isomorphism (2.2.4) induces an action of Γ ( c l Loc c G , Q Q , O ) Γ c l Loc c ⁡ G , Q ⊗ Q â„“ , O Gamma(^(cl)Loc^(c)_(G,Q)oxQ_(â„“),O)\Gamma\left({ }^{c l} \operatorname{Loc}^{c}{ }_{G, Q} \otimes \mathbb{Q}_{\ell}, \mathcal{O}\right)Γ(clLocc⁡G,Q⊗Qâ„“,O) on the righthand side. This is exactly the action by V V V\mathrm{V}V. Lafforgue's excursion operators, which induces the decomposition of the right-hand side (in particular, C c ( G ( F ) G ( A F ) / K , Q ) C c G ( F ) ∖ G A F / K , Q â„“ C_(c)(G(F)\\G(A_(F))//K,Q_(â„“))C_{c}\left(G(F) \backslash G\left(\mathbb{A}_{F}\right) / K, \mathbb{Q}_{\ell}\right)Cc(G(F)∖G(AF)/K,Qâ„“) ) in terms of semisimple Langlands parameters. As explained [40], over an elliptic Langlands parameter, such an isomorphism is closely related to the Arthur-Kottwitz multiplicity formula. In the case of G = G L n G = G L n G=GL_(n)G=\mathrm{GL}_{n}G=GLn, it gives the following corollary, generalizing [38].
Corollary 2.2.5. Let π Ï€ pi\piÏ€ be a cuspidal automorphic representation of G L n G L n GL_(n)\mathrm{GL}_{n}GLn, with the associated irreducible Galois representation ρ π : W F , Q GL n ( Λ ) ρ Ï€ : W F , Q → GL n ⁡ ( Λ ) rho_(pi):W_(F,Q)rarrGL_(n)(Lambda)\rho_{\pi}: W_{F, Q} \rightarrow \operatorname{GL}_{n}(\Lambda)ρπ:WF,Q→GLn⁡(Λ) for some finite extension Λ / Q Λ / Q â„“ Lambda//Q_(â„“)\Lambda / \mathbb{Q}_{\ell}Λ/Qâ„“ and with m π m Ï€ m_(pi)\mathfrak{m}_{\pi}mÏ€ the corresponding maximal ideal of Γ ( c l Loc G , Q Λ , O ) Γ c l Loc G , Q ⊗ Λ , O Gamma(^(cl)Loc^(G),Q ox Lambda,O)\Gamma\left({ }^{c l} \operatorname{Loc}^{G}, Q \otimes \Lambda, \mathcal{O}\right)Γ(clLocG,Q⊗Λ,O). Then there is an ( W F , Q S × H K ) W F , Q S × H K (W_(F,Q)^(S)xxH_(K))\left(W_{F, Q}^{S} \times H_{K}\right)(WF,QS×HK)-equivariant isomorphism
H c ( Sht K ( G ) Δ ( η ¯ ) , Sat S ( V ) ) / m π V ρ π π K H c ∗ Sht K ⁡ ( G ) Δ ( η ¯ ) , Sat S ⁡ ( V ) / m Ï€ ≅ V ρ Ï€ ⊗ Ï€ K H_(c)^(**)(Sht_(K)(G)_(Delta( bar(eta))),Sat_(S)(V))//m_(pi)~=V_(rho_(pi))oxpi^(K)H_{c}^{*}\left(\operatorname{Sht}_{K}(G)_{\Delta(\bar{\eta})}, \operatorname{Sat}_{S}(V)\right) / \mathfrak{m}_{\pi} \cong V_{\rho_{\pi}} \otimes \pi^{K}Hc∗(ShtK⁡(G)Δ(η¯),SatS⁡(V))/mπ≅Vρπ⊗πK
In particular, the left-hand side only concentrates in cohomological degree zero.

2.3. Geometric realization of Jacquet-Langlands transfer

The global Langlands correspondence for number fields is far more complicated In fact, there are analytic part of the theory which currently seems not to fit the categorification/decategorification framework. Even if we just restrict to the algebraic/arithmetic part of the theory, there are complications coming from the place at ℓ ℓ\ellℓ and at ∞ oo\infty∞. In particular, the categorical forms of the local Langlands correspondence at ℓ ℓ\ellℓ and ∞ oo\infty∞ are not yet fully understood.
Nevertheless, in a forthcoming joint work with Emerton and Emerton-Gee [21, 22], we will formulate conjectural Galois theoretical descriptions for the cohomology of Shimura varieties and even cohomology for general locally symmetric space, parallel to Conjecture 2.2.3. In this subsection, we just review a conjecture from [82] on the geometric realization of Jacquet-Langlands transfer via cohomology of Shimura varieties and discuss results from [ 35 , 64 ] [ 35 , 64 ] [35,64][35,64][35,64] towards this conjecture.
We fix a few notations and assumptions. We fix a prime p p ppp in this subsection.
Let A f = q Q q A f = ∏ q ′   Q q A_(f)=prod_(q)^(')Q_(q)\mathbb{A}_{f}=\prod_{q}^{\prime} \mathbb{Q}_{q}Af=∏q′Qq denote the ring of finite adèles of Q Q Q\mathbb{Q}Q, and A f p = q p Q p A f p = ∏ q ≠ p ′   Q p A_(f)^(p)=prod_(q!=p)^(')Q_(p)\mathbb{A}_{f}^{p}=\prod_{q \neq p}^{\prime} \mathbb{Q}_{p}Afp=∏q≠p′Qp. We write η ¯ = Spec Q ¯ η ¯ = Spec ⁡ Q ¯ bar(eta)=Spec bar(Q)\bar{\eta}=\operatorname{Spec} \overline{\mathbb{Q}}η¯=Spec⁡Q¯, where Q ¯ Q ¯ bar(Q)\overline{\mathbb{Q}}Q¯ is the algebraic closure of Q Q Q\mathbb{Q}Q in C C C\mathbb{C}C. For a Shimura datum ( G , X ) ( G , X ) (G,X)(G, X)(G,X), let μ μ mu\muμ be the (minuscule) dominant weight of G ^ G ^ hat(G)\hat{G}G^ (with respect to ( B ^ , T ^ ) ( B ^ , T ^ ) ( hat(B), hat(T))(\hat{B}, \hat{T})(B^,T^) ) determined by ( G , X ) ( G , X ) (G,X)(G, X)(G,X) in the usual way and let V μ V μ V_(mu)V_{\mu}Vμ denote the minuscule representation of G ^ G ^ hat(G)\hat{G}G^ of highest weight μ μ mu\muμ. Let E Q ¯ C E ⊂ Q ¯ ⊂ C E sub bar(Q)subCE \subset \overline{\mathbb{Q}} \subset \mathbb{C}E⊂Q¯⊂C be the reflex field of ( G , X ) ( G , X ) (G,X)(G, X)(G,X) and write d μ = dim X d μ = dim ⁡ X d_(mu)=dim Xd_{\mu}=\operatorname{dim} Xdμ=dim⁡X. For a level (i.e., an open compact subgroup) K = K p K p G ( Q p ) G ( A f p ) K = K p K p ⊂ G Q p G A f p K=K_(p)K^(p)sub G(Q_(p))G(A_(f)^(p))K=K_{p} K^{p} \subset G\left(\mathbb{Q}_{p}\right) G\left(\mathbb{A}_{f}^{p}\right)K=KpKp⊂G(Qp)G(Afp), let Sh K ( G ) Sh K ⁡ ( G ) Sh_(K)(G)\operatorname{Sh}_{K}(G)ShK⁡(G) be the corresponding
Shimura variety of level K K KKK (defined over the reflex field E E EEE ), and let Sh K ( G ) η ¯ Sh K ⁡ ( G ) η ¯ Sh_(K)(G)_( bar(eta))\operatorname{Sh}_{K}(G)_{\bar{\eta}}ShK⁡(G)η¯ denote its base change along E Q ¯ E → Q ¯ E rarr bar(Q)E \rightarrow \overline{\mathbb{Q}}E→Q¯. Let v v vvv be a place of E E EEE above p p ppp. By a specialization s p : η ¯ v ¯ s p : η ¯ → v ¯ sp: bar(eta)rarr bar(v)\mathrm{sp}: \bar{\eta} \rightarrow \bar{v}sp:η¯→v¯, we mean a morphism from η ¯ η ¯ bar(eta)\bar{\eta}η¯ to the strict henselianization of O E O E O_(E)\mathcal{O}_{E}OE at v v vvv.
To avoid many complications from Galois cohomology (e.g., the difference between extended pure inner forms and inner forms) and also some complications from geometry (e.g., the relation between Shimura varieties and moduli of Shtukas), we assume that G G GGG is of adjoint type in the rest of this subsection, and refer to [64] for general G G GGG. See also [82] with less restrictions on G G GGG.
Definition 2.3.1. Let G G GGG be a connected reductive group over Q Q Q\mathbb{Q}Q. A prime-to- p p ppp (resp. finitely) trivialized inner form of G G GGG is a G G GGG-torsor β β beta\betaβ over Q Q Q\mathbb{Q}Q equipped with a trivialization β β beta\betaβ over A f p A f p A_(f)^(p)\mathbb{A}_{f}^{p}Afp (resp. over A f A f A_(f)\mathbb{A}_{f}Af ). Then G := Aut ( ξ ) G ′ := Aut ⁡ ( ξ ) G^('):=Aut(xi)G^{\prime}:=\operatorname{Aut}(\xi)G′:=Aut⁡(ξ) is an inner form of G G GGG (so the dual group of G G GGG and G G ′ G^(')G^{\prime}G′ are canonically identified), equipped with an isomorphism θ : G ( A f p ) G ( A f p ) θ : G A f p ≅ G ′ A f p theta:G(A_(f)^(p))~=G^(')(A_(f)^(p))\theta: G\left(\mathbb{A}_{f}^{p}\right) \cong G^{\prime}\left(\mathbb{A}_{f}^{p}\right)θ:G(Afp)≅G′(Afp) (resp. θ : G ( A f ) G ( A f ) ) θ : G A f ≅ G ′ A f {: theta:G(A_(f))~=G^(')(A_(f)))\left.\theta: G\left(\mathbb{A}_{f}\right) \cong G^{\prime}\left(\mathbb{A}_{f}\right)\right)θ:G(Af)≅G′(Af)).
Now let ( G , X ) ( G , X ) (G,X)(G, X)(G,X) and ( G , X ) G ′ , X ′ (G^('),X^('))\left(G^{\prime}, X^{\prime}\right)(G′,X′) be two Shimura data, with G G ′ G^(')G^{\prime}G′ a prime-to- p p ppp trivialized inner form of G G GGG. Via θ θ theta\thetaθ, one can transport K p G ( A f p ) K p ⊂ G A f p K^(p)sub G(A_(f)^(p))K^{p} \subset G\left(\mathbb{A}_{f}^{p}\right)Kp⊂G(Afp) to an open compact subgroup K p G ( A f p ) K ′ p ⊂ G ′ A f p K^('p)subG^(')(A_(f)^(p))K^{\prime p} \subset G^{\prime}\left(\mathbb{A}_{f}^{p}\right)K′p⊂G′(Afp). We identify the prime-to- p p ppp (derived) Hecke algebra H K p , Λ H K p , Λ H_(K^(p),Lambda)H_{K^{p}, \Lambda}HKp,Λ (defined in the same way as in (2.5)) with H K p , Λ H K ′ p , Λ H_(K^('p),Lambda)H_{K^{\prime p}, \Lambda}HK′p,Λ and simply write them as H K p , Λ H K p , Λ H_(K^(p),Lambda)H_{K^{p}, \Lambda}HKp,Λ. Let K p G ( Q p ) K p ′ ⊂ G ′ Q p K_(p)^(')subG^(')(Q_(p))K_{p}^{\prime} \subset G^{\prime}\left(\mathbb{Q}_{p}\right)Kp′⊂G′(Qp) be an open compact subgroup and write K = K p K p K ′ = K p ′ K ′ p K^(')=K_(p)^(')K^('p)K^{\prime}=K_{p}^{\prime} K^{\prime p}K′=Kp′K′p for the corresponding level.
We fix a quasisplit inner form G Q p G Q p ∗ G_(Q_(p))^(**)G_{\mathbb{Q}_{p}}^{*}GQp∗ of G Q p G Q p G_(Q_(p))G_{\mathbb{Q}_{p}}GQp and G Q p G Q p ′ G_(Q_(p))^(')G_{\mathbb{Q}_{p}}^{\prime}GQp′ equipped with a pinning ( B Q p , T Q p , e ) B Q p ∗ , T Q p ∗ , e ∗ (B_(Q_(p))^(**),T_(Q_(p))^(**),e^(**))\left(B_{\mathbb{Q}_{p}}^{*}, T_{\mathbb{Q}_{p}}^{*}, e^{*}\right)(BQp∗,TQp∗,e∗), and realize G Q p G Q p G_(Q_(p))G_{\mathbb{Q}_{p}}GQp as J b J b J_(b)J_{b}Jb and G Q p G Q p ′ G_(Q_(p))^(')G_{\mathbb{Q}_{p}}^{\prime}GQp′ as J b J b ′ J_(b^('))J_{b^{\prime}}Jb′ for b , b B ( G Q p ) b , b ∈ B G Q p ∗ b,b in B(G_(Q_(p))^(**))b, b \in B\left(G_{\mathbb{Q}_{p}}^{*}\right)b,b∈B(GQp∗). Under our assumption that G G GGG and G G ′ G^(')G^{\prime}G′ are adjoint, such b , b b , b ′ b,b^(')b, b^{\prime}b,b′ exist and are unique. Then we have the conjectural coherent sheaf U K p , Λ U K p , Λ U_(K_(p),Lambda)\mathfrak{U}_{K_{p}, \Lambda}UKp,Λ and H K p , Λ H K p ′ , Λ H_(K_(p)^('),Lambda)\mathscr{H}_{K_{p}^{\prime}, \Lambda}HKp′,Λ as in (2.1) on the stack Loc G , p Λ G , p ⊗ Λ _(G,p)ox Lambda{ }_{G, p} \otimes \LambdaG,p⊗Λ of local Langlands parameters for G Q p G Q p ∗ G_(Q_(p))^(**)G_{\mathbb{Q}_{p}}^{*}GQp∗ over Λ Î› Lambda\LambdaΛ.
Conjecture 2.3.2. For every choice of specialization map s p : η ¯ v ¯ s p : η ¯ → v ¯ sp: bar(eta)rarr bar(v)\mathrm{sp}: \bar{\eta} \rightarrow \bar{v}sp:η¯→v¯, there is a natural map
R Hom Coh ( Locc G , p Λ ) ( V μ ~ N K p , Λ , V μ ~ N K p , Λ ) (2.6) R Hom H K p , Λ ( C c ( Sh K ( G ) η ¯ , Λ [ d μ ] ) , C c ( Sh K ( G ) η ¯ , Λ [ d μ ] ) ) R Hom Coh ⁡ Locc G , p ⊗ Λ ⁡ V μ ~ ⊗ N K p , Λ , V μ ′ ~ ⊗ N K p ′ , Λ (2.6) → R Hom H K p , Λ ⁡ C c Sh K ⁡ ( G ) η ¯ , Λ d μ , C c Sh K ′ ⁡ G ′ η ¯ , Λ d μ ′ {:[RHom_(Coh(Locc_(G,p)ox Lambda))(( widetilde(V_(mu)))oxN_(K_(p),Lambda),( widetilde(V_(mu^('))))oxN_(K_(p)^('),Lambda))],[(2.6)quad rarr RHom_(H_(K)p,Lambda)(C_(c)(Sh_(K)(G)_( bar(eta)),Lambda[d_(mu)]),C_(c)(Sh_(K^('))(G^('))_( bar(eta)),Lambda[d_(mu^('))]))]:}\begin{align*} & R \operatorname{Hom}_{\operatorname{Coh}\left(\operatorname{Locc}_{G, p} \otimes \Lambda\right)}\left(\widetilde{V_{\mu}} \otimes \mathfrak{N}_{K_{p}, \Lambda}, \widetilde{V_{\mu^{\prime}}} \otimes \mathfrak{N}_{K_{p}^{\prime}, \Lambda}\right) \\ & \quad \rightarrow R \operatorname{Hom}_{H_{K} p, \Lambda}\left(C_{c}\left(\operatorname{Sh}_{K}(G)_{\bar{\eta}}, \Lambda\left[d_{\mu}\right]\right), C_{c}\left(\operatorname{Sh}_{K^{\prime}}\left(G^{\prime}\right)_{\bar{\eta}}, \Lambda\left[d_{\mu^{\prime}}\right]\right)\right) \tag{2.6} \end{align*}RHomCoh⁡(LoccG,p⊗Λ)⁡(Vμ~⊗NKp,Λ,Vμ′~⊗NKp′,Λ)(2.6)→RHomHKp,Λ⁡(Cc(ShK⁡(G)η¯,Λ[dμ]),Cc(ShK′⁡(G′)η¯,Λ[dμ′]))
compatible with compositions. In particular, there is an ( E 1 E 1 − E_(1^(-))E_{1^{-}}E1−)algebra homomorphism
compatible with (2.6). In addition, the induced action
(2.8) H K p , Λ ( 2.2 ) R End ( U K p , Λ ) R End ( V μ ~ U K p , Λ ) S R End H K p , Λ ( C c ( Sh K ( G ) η ¯ , Λ ) ) (2.8) H K p , Λ ≅ ( 2.2 ) R End ⁡ U K p , Λ → R End ⁡ V μ ~ ⊗ U K p , Λ → S R End H K p , Λ ⁡ C c Sh K ⁡ ( G ) η ¯ , Λ {:(2.8)H_(K_(p),Lambda)~=^((2.2))R End(U_(K_(p),Lambda))rarr R End(( widetilde(V_(mu)))oxU_(K_(p),Lambda))rarr"S"REnd_(H_(K^(p),Lambda))(C_(c)(Sh_(K)(G)_( bar(eta)),Lambda)):}\begin{equation*} H_{K_{p}, \Lambda} \stackrel{(2.2)}{\cong} R \operatorname{End}\left(\mathfrak{U}_{K_{p}, \Lambda}\right) \rightarrow R \operatorname{End}\left(\widetilde{V_{\mu}} \otimes \mathfrak{U}_{K_{p}, \Lambda}\right) \xrightarrow{S} R \operatorname{End}_{H_{K^{p}, \Lambda}}\left(C_{c}\left(\operatorname{Sh}_{K}(G)_{\bar{\eta}}, \Lambda\right)\right) \tag{2.8} \end{equation*}(2.8)HKp,Λ≅(2.2)REnd⁡(UKp,Λ)→REnd⁡(Vμ~⊗UKp,Λ)→SREndHKp,Λ⁡(Cc(ShK⁡(G)η¯,Λ))
coincides with the natural Hecke action of H K p , Λ H K p , Λ H_(K_(p),Lambda)H_{K_{p}, \Lambda}HKp,Λ on C c ( Sh K ( G ) η ¯ , Λ ) C c Sh K ⁡ ( G ) η ¯ , Λ C_(c)(Sh_(K)(G)_( bar(eta)),Lambda)C_{c}\left(\operatorname{Sh}_{K}(G)_{\bar{\eta}}, \Lambda\right)Cc(ShK⁡(G)η¯,Λ) (and therefore is independent of the specialization map s p s p sp\mathrm{sp}sp ).
This conjecture would be a consequence of a Galois theoretic description of C c ( Sh K ( G ) η ¯ , Λ ) C c Sh K ⁡ ( G ) η ¯ , Λ C_(c)(Sh_(K)(G)_( bar(eta)),Lambda)C_{c}\left(\operatorname{Sh}_{K}(G)_{\bar{\eta}}, \Lambda\right)Cc(ShK⁡(G)η¯,Λ) similar to Conjecture 2.2.3, but its formulation does not require the existence of the stack of global Langlands parameters for Q Q Q\mathbb{Q}Q. In any case, a step towards a Galois-theoretical description of C c ( Sh K ( G ) η ¯ , Λ ) C c Sh K ⁡ ( G ) η ¯ , Λ C_(c)(Sh_(K)(G)_( bar(eta)),Lambda)C_{c}\left(\operatorname{Sh}_{K}(G)_{\bar{\eta}}, \Lambda\right)Cc(ShK⁡(G)η¯,Λ) might require Conjecture 2.3.2 as an
input. We also remark that as in the function field case, there is a more general version of such conjecture in [82, SECT. 4.7], allowing "generalized level structures," so that the cohomology of Igusa varieties could appear.
The following theorem verifies the conjecture in special cases.
Theorem 2.3.3. Suppose that the Shimura data ( G , X ) ( G , X ) (G,X)(G, X)(G,X) and ( G , X ) G ′ , X ′ (G^('),X^('))\left(G^{\prime}, X^{\prime}\right)(G′,X′) are of abelian type, with G G ′ G^(')G^{\prime}G′ a finitely trivialized inner form of G G GGG. Suppose that G Q p G Q p G_(Q_(p))G_{\mathbb{Q}_{p}}GQp is unramified (and therefore so is G Q p G Q p ′ G_(Q_(p))^(')G_{\mathbb{Q}_{p}}^{\prime}GQp′ ).
(1) The map (2.6) (and therefore (2.7)) exists when Λ = Q Λ = Q â„“ Lambda=Q_(â„“)\Lambda=\mathbb{Q}_{\ell}Λ=Qâ„“ and K p G ( Q p ) K p ⊂ G Q p K_(p)sub G(Q_(p))K_{p} \subset G\left(\mathbb{Q}_{p}\right)Kp⊂G(Qp) and K p G ( Q p ) K p ′ ⊂ G ′ Q p K_(p)^(')subG^(')(Q_(p))K_{p}^{\prime} \subset G^{\prime}\left(\mathbb{Q}_{p}\right)Kp′⊂G′(Qp) are parahoric subgroups (in the sense of Bruhat-Tits).
(2) If K p K p K_(p)K_{p}Kp is hyperspecial, then the map (2.6) (and therefore (2.7)) exists when Λ = Z Λ = Z â„“ Lambda=Z_(â„“)\Lambda=\mathbb{Z}_{\ell}Λ=Zâ„“, at least for underived Hom spaces. In addition, the action of H K p c l H K p c l H_(K_(p))^(cl)H_{K_{p}}^{\mathrm{cl}}HKpcl on H c ( Sh K ( G ) η ¯ , Λ ) H c ∗ Sh K ⁡ ( G ) η ¯ , Λ H_(c)^(**)(Sh_(K)(G)_( bar(eta)),Lambda)H_{c}^{*}\left(\operatorname{Sh}_{K}(G)_{\bar{\eta}}, \Lambda\right)Hc∗(ShK⁡(G)η¯,Λ) via (2.8) coincides with the natural action of H K p c l H K p c l H_(K_(p))^(cl)H_{K_{p}}^{\mathrm{cl}}HKpcl.
Part (1) is proved in [35,64]. The proof contains two ingredients. One is the construction of physical correspondences between mod p p ppp fibers of S h K ( G ) S h K ( G ) Sh_(K)(G)\mathrm{Sh}_{K}(G)ShK(G) and S h K ( G ) S h K ′ G ′ Sh_(K^('))(G^('))\mathrm{Sh}_{K^{\prime}}\left(G^{\prime}\right)ShK′(G′) by [64] (this is where we currently need to assume that G G GGG and G G ′ G^(')G^{\prime}G′ are unramified at p p ppp ). The other ingredient is Theorem 2.1.5 (and therefore requires Λ = Q Λ = Q â„“ Lambda=Q_(â„“)\Lambda=\mathbb{Q}_{\ell}Λ=Qâ„“ ). When K p K p K_(p)K_{p}Kp is hypersepcial, one can work with Z Z â„“ Z_(â„“)\mathbb{Z}_{\ell}Zâ„“-coefficient, as (the underived version of) (2.4) exists for Z Z â„“ Z_(â„“)\mathbb{Z}_{\ell}Zâ„“-coefficient thanks to [70]. In fact, in this case one can allow nontrivial local systems on the Shimura varieties (see [70]). The last statement is known as the S = T S = T S=TS=TS=T for Shimura varieties. The case when d μ = dim Sh K ( G ) = 0 d μ = dim ⁡ Sh K ⁡ ( G ) = 0 d_(mu)=dim Sh_(K)(G)=0d_{\mu}=\operatorname{dim} \operatorname{Sh}_{K}(G)=0dμ=dim⁡ShK⁡(G)=0 is contained in [64]. The general case is proved in [63,74] using foundational works from [ 25 , 59 ] [ 25 , 59 ] [25,59][25,59][25,59].

3. APPLICATIONS TO ARITHMETIC GEOMETRY

Besides the previously mentioned directly applications of (ideas from) geometric Langlands to the classical Langlands program, we discuss some further arithmetic applications, mostly related to Shimura varieties and based on the author's works. We shall mention that there are many other remarkable applications of (ideas of) geometric Langlands to arithmetic problems, such as [ 28 , 31 , 44 , 66 , 71 ] [ 28 , 31 , 44 , 66 , 71 ] [28,31,44,66,71][28,31,44,66,71][28,31,44,66,71], to name a few.

3.1. Local models of Shimura varieties

The theory of integral models of Shimura varieties (with parahoric level) started (implicitly in the work of Kronecker) with understanding of the mod p mod p mod p\bmod pmodp reduction of elliptic modular curves with Γ 0 ( p ) Γ 0 ( p ) Gamma_(0)(p)\Gamma_{0}(p)Γ0(p)-level. We discuss a small fraction of this theory concerning étale local structures of these integral models via the theory of local models. The recent developments of the theory of local models are greatly influenced by the geometric Langlands program.
We use notations from Section 2.3 for Shimura varieties (but we do not assume that G G GGG is of adjoint type in this subsection). Let ( G , X ) ( G , X ) (G,X)(G, X)(G,X) be a Shimura datum and K K KKK a chosen level with K p = E ( Z p ) K p = E Z p K_(p)=E(Z_(p))K_{p}=\mathscr{E}\left(\mathbb{Z}_{p}\right)Kp=E(Zp) for some parahoric group scheme E E E\mathscr{E}E (in the sense of Bruhat-Tits) of
G Q p G Q p G_(Q_(p))G_{\mathbb{Q}_{p}}GQp over Z p Z p Z_(p)\mathbb{Z}_{p}Zp. Then for a place v v vvv of E E EEE over p p ppp, a local model diagram is a correspondence of quasiprojective schemes over O E v O E v O_(E_(v))\mathcal{O}_{E_{v}}OEv,
(3.1) S K S ~ K φ ~ M G l o c (3.1) S K ← S ~ K → φ ~ M G l o c {:(3.1)S_(K)larr tilde(S)_(K)rarr" tilde(varphi)"M_(G)^(loc):}\begin{equation*} \mathscr{S}_{K} \leftarrow \tilde{\mathscr{S}}_{K} \xrightarrow{\tilde{\varphi}} M_{\mathscr{G}}^{\mathrm{loc}} \tag{3.1} \end{equation*}(3.1)SK←S~K→φ~MGloc
where S K S K S_(K)\mathscr{S}_{K}SK is an integral model of Sh K ( G ) Sh K ⁡ ( G ) Sh_(K)(G)\operatorname{Sh}_{K}(G)ShK⁡(G) over O E v , S ~ K O E v , S ~ K O_(E_(v)), tilde(S)_(K)\mathcal{O}_{E_{v}}, \tilde{\mathscr{S}}_{K}OEv,S~K is a E O E v E O E v E_(O_(E_(v)))\mathscr{E}_{\mathcal{O}_{E_{v}}}EOEv-torsor over S K , M G loc S K , M G loc  S_(K),M_(G)^("loc ")\mathscr{S}_{K}, M_{\mathscr{G}}^{\text {loc }}SK,MGloc  is the so-called local model, which is a flat projective scheme over O E v O E v O_(E_(v))\mathcal{O}_{E_{v}}OEv equipped with a E O E v E O E v E_(O_(E_(v)))\mathscr{E}_{\mathcal{O}_{E_{v}}}EOEv-action, and φ ~ φ ~ tilde(varphi)\tilde{\varphi}φ~ is a E O E v E O E v E_(O_(E_(v)))\mathscr{E}_{\mathcal{O}_{E_{v}}}EOEv-equivariant smooth morphism of relative dimension dim G G GGG Therefore, M G loc M G loc  M_(G)^("loc ")M_{\mathscr{G}}^{\text {loc }}MGloc  models étale local structure of S K S K S_(K)\mathscr{S}_{K}SK. On the other hand, the existence of U O E v U O E v U_(O_(E_(v)))\mathscr{U}_{\mathcal{O}_{E_{v}}}UOEv-action on M G loc M G loc  M_(G)^("loc ")M_{\mathscr{G}}^{\text {loc }}MGloc  makes it easier than S K S K S_(K)\mathscr{S}_{K}SK to study.
The original construction of local models is based on realization of a parahoric group scheme as (the neutral connected component of) the stabilizer group of a self-dual lattice chain in a vector space (over a division algebra over F F FFF ) with a bilinear form, e.g., see [57] for a survey and references. This approach is somehow ad hoc and is limited the so-called (P)EL (local) Shimura data. A new approach, based on the construction of an Z p Z p Z_(p)\mathbb{Z}_{p}Zp-analogue of the stack H k g , D H k g , D Hk_(g,D)\mathrm{Hk}_{\boldsymbol{g}, D}Hkg,D from Section 1.2, was systematically introduced in [58] (under the tameness assumption of G G GGG which was later lifted in [46,50]). In [58] the construction of such a Z p Z p Z_(p)\mathbb{Z}_{p}Zp-analogue (or rather the corresponding Beilinson-Drinfeld-type affine Grassmannian Gr Z p Gr ⁡ Z p Gr Z_(p)\operatorname{Gr} \mathscr{Z}_{\mathbb{p}}Gr⁡Zp over Z p Z p Z_(p)\mathbb{Z}_{p}Zp ) is based on the construction of certain "two dimensional parahoric" group scheme E ~ E ~ tilde(E)\tilde{\mathscr{E}}E~ over Z p [ ϖ ] Z p [ Ï– ] Z_(p)[Ï–]\mathbb{Z}_{p}[\varpi]Zp[Ï–] whose restriction along Z p [ ϖ ] ϖ p Z p Z p [ Ï– ] → Ï– ↦ p Z p Z_(p)[Ï–]rarr"Ï–|->p"Z_(p)\mathbb{Z}_{p}[\varpi] \xrightarrow{\varpi \mapsto p} \mathbb{Z}_{p}Zp[Ï–]→ϖ↦pZp recovers E E E\mathscr{E}E. (See [81] for a survey.) A more direct construction of a different p p ppp-adic version of such affine Grassman-

model is defined as the flat closure of the Schubert variety in the generic fiber corresponding to μ μ mu\muμ. In addition, the recent work [1] shows that the two constructions agree. The following theorem from [1] is the most up-to-date result on the existence of local models and about their properties.
Theorem 3.1.1. Let G G GGG be a connected reductive group over a p-adic field F F FFF. Except the odd unitary case when p = 2 p = 2 p=2p=2p=2 and triality case when p = 3 p = 3 p=3p=3p=3, for every parahoric group scheme E E E\mathcal{E}E of G G GGG over O O O\mathcal{O}O, and a conjugacy class of minuscule cocharacters μ μ mu\muμ of G G GGG defined over a finite extension E / F E / F E//FE / FE/F of F F FFF, there is a normal flat projective scheme M G , μ l o c M G , μ l o c M_(G,mu)^(loc)M_{\mathscr{G}, \mu}^{\mathrm{loc}}MG,μloc over O E O E O_(E)\mathcal{O}_{E}OE, equipped with a E O E E O E E_(O_(E))\mathscr{E}_{\mathcal{O}_{E}}EOE-action such that M Q , μ l o c E M Q , μ l o c ⊗ E M_(Q,mu)^(loc)ox EM_{\mathscr{Q}, \mu}^{\mathrm{loc}} \otimes EMQ,μloc⊗E is G E G E G_(E)G_{E}GE-equivariantly isomorphic to the partial flag variety F μ F â„“ μ Fâ„“_(mu)\mathscr{F} \ell_{\mu}Fℓμ of G E G E G_(E)G_{E}GE corresponding to μ μ mu\muμ, and that M G l o c k E M G l o c ⊗ k E M_(G)^(loc)oxk_(E)M_{\mathscr{G}}^{\mathrm{loc}} \otimes k_{E}MGloc⊗kE is ( G k E ) G ⊗ k E (Goxk_(E))\left(\mathscr{G} \otimes k_{E}\right)(G⊗kE)-equivariantly isomorphic to the (canonical deperfection of the) union over the μ μ mu\muμ-admissible set of Schubert varieties in L G / L + E k E L G / L + E ⊗ k E LG//L^(+)Eoxk_(E)L G / L^{+} \mathcal{E} \otimes k_{E}LG/L+E⊗kE. In addition, M G loc M G loc  M_(G)^("loc ")M_{\mathscr{G}}^{\text {loc }}MGloc  is normal, Cohen-Macaulay and each of its geometric irreducible components in its special fiber is normal and Cohen-Macaulay.
We end this subsection with a few remarks.
Remark 3.1.2. (1) Once the local model diagram (3.1) is established, this theorem also gives the corresponding properties of the integral models of Shimura varieties.
(2) A key ingredient in the study of special fibers of local models is the coherence conjecture by Pappas-Rapoport [56], proved in [75] (and the proof uses the idea of fusion).
(3) One important motivation/application of the theory of local models is the Haines-Kottwitz conjecture [29], which predicts certain central element in the parahoric Hecke algebra H K p c l H K p c l H_(K_(p))^(cl)H_{K_{p}}^{\mathrm{cl}}HKpcl should be used as the test function in the trace formula computing the Hasse-Weil zeta function of S h K ( G ) S h K ( G ) Sh_(K)(G)\mathrm{Sh}_{K}(G)ShK(G). As mentioned in Section 1.2, this conjecture motivated Gaitsgory's central sheaf construction (1.9). With the local Hecke stack H k Z , Z p H k Z , Z p HkZ_(,)Z_(p)\mathrm{Hk} \mathscr{Z}_{,} \mathbb{Z}_{p}HkZ,Zp over Z p Z p Z_(p)\mathbb{Z}_{p}Zp constructed (either the version from [58] or from [59]), one can mimic the construction (1.9) in mixed characteristic to solve the Kottwitz conjecture. Again, see [1] for the up-to-date result.

3.2. The congruence relation

We use notations and (for simplicity) keep assumptions from Section 2.3 regarding Shimura varieties. Let ( G , X ) ( G , X ) (G,X)(G, X)(G,X) be a Shimura datum abelian type, and let K K KKK be a level such that K p K p K_(p)K_{p}Kp is hyperspecial. Let v p v ∣ p v∣pv \mid pv∣p be the place of E E EEE. Then Sh K ( G ) Sh K ⁡ ( G ) Sh_(K)(G)\operatorname{Sh}_{K}(G)ShK⁡(G) has a canonical integral model S K S K S_(K)\mathscr{S}_{K}SK defined over O E , ( v ) O E , ( v ) O_(E,(v))\mathcal{O}_{E,(v)}OE,(v) [37]. Let S ¯ K S ¯ K bar(S)_(K)\overline{\mathscr{S}}_{K}S¯K be its mod p p ppp fiber, which is a smooth variety defined over the residue field k v k v k_(v)k_{v}kv of v v vvv. Let σ v σ v sigma_(v)\sigma_{v}σv denote the geometric Frobenius in Γ k v Γ k v Gamma_(k_(v))\Gamma_{k_{v}}Γkv. Theorem 2.3.3

consequences.
The congruence relation conjecture (also known as the Blasius-Rogawski conjecture), generalizing the classical Eichler-Shimura congruence relation Frob p = T p + V p p = T p + V p _(p)=T_(p)+V_(p){ }_{p}=T_{p}+V_{p}p=Tp+Vp for modular curves, predicts that in the Chow group of S ¯ K × S ¯ K S ¯ K × S ¯ K bar(S)_(K)xx bar(S)_(K)\overline{\mathscr{S}}_{K} \times \overline{\mathscr{S}}_{K}S¯K×S¯K, the Frobenius endomorphism of S ¯ K S ¯ K bar(S)_(K)\overline{\mathscr{S}}_{K}S¯K satisfies a polynomial whose coefficients are mod p p ppp reduction of certain Hecke correspondences. Theorem 2.3.3, together with [65, SECT. 6.3], implies this conjecture at the level of cohomology.
For every representation V V VVV of c ( G Q p ) c G Q p ^(c)(G_(Q_(p))){ }^{c}\left(G_{\mathbb{Q}_{p}}\right)c(GQp), its character χ V χ V chi_(V)\chi_{V}χV (regarded as a G ^ G ^ hat(G)\hat{G}G^-invariant function on c G | d = ( p , σ p ) ) c G d = p , σ p {:^(c)G|_(d=(p,sigma_(p))))\left.\left.{ }^{c} G\right|_{d=\left(p, \sigma_{p}\right)}\right)cG|d=(p,σp)) gives an element h V H G ( Z p ) c l h V ∈ H G Z p c l h_(V)inH_(G(Z_(p)))^(cl)h_{V} \in H_{G\left(\mathbb{Z}_{p}\right)}^{\mathrm{cl}}hV∈HG(Zp)cl via the Satake isomorphism (1.3).
Theorem 3.2.1. The following identity,
(3.2) i = 0 n ( 1 ) j h χ j V σ v dim V j = 0 (3.2) ∑ i = 0 n   ( − 1 ) j h χ ∧ j V σ v dim ⁡ V − j = 0 {:(3.2)sum_(i=0)^(n)(-1)^(j)h_(chi_(^^)j_(V))sigma_(v)^(dim V-j)=0:}\begin{equation*} \sum_{i=0}^{n}(-1)^{j} h_{\chi_{\wedge} j_{V}} \sigma_{v}^{\operatorname{dim} V-j}=0 \tag{3.2} \end{equation*}(3.2)∑i=0n(−1)jhχ∧jVσvdim⁡V−j=0
Indeed, by [65, sEct. 6.3], such an equality holds with h χ i V h χ ∧ i V h_(chi_(^^^(i)V))h_{\chi_{\wedge^{i} V}}hχ∧iV replaced by S ( χ i V ) S χ ∧ i V S(chi_(^^^(i)V))S\left(\chi_{\wedge^{i} V}\right)S(χ∧iV), where S S SSS is from Theorem 2.3.3 (1). Then part (2) of that theorem allows one to replace S ( χ i V ) S χ ∧ i V S(chi_(^^^(i)V))S\left(\chi_{\wedge^{i} V}\right)S(χ∧iV) by h χ i V h χ ∧ i V h_(chi_(^^^(i)V))h_{\chi_{\wedge^{i} V}}hχ∧iV. This approach to (3.2) is the Shimura variety analogue of V. Lafforgue's approach to the Eichler-Shimura relation for Sht K ( G ) Sht K ⁡ ( G ) Sht_(K)(G)\operatorname{Sht}_{K}(G)ShtK⁡(G) [39]. Traditionally, there is another approach to the congruence relation conjecture for Shimura varieties by directly studying reduction mod p p ppp of Hecke operators, starting from [24] for the Siegel modular variety case.
See [45] for the latest progress and related references. This approach would give (3.2) at the level of algebraic correspondences.
Now suppose ( G , X ) = ( Res F + / Q ( G 0 ) F + , φ : F + R X 0 ) ( G , X ) = Res F + / Q ⁡ G 0 F + , ∏ φ : F + → R   X 0 (G,X)=(Res_(F^(+)//Q)(G_(0))_(F^(+)),prod_(varphi:F^(+)rarrR)X_(0))(G, X)=\left(\operatorname{Res}_{F^{+} / \mathbb{Q}}\left(G_{0}\right)_{F^{+}}, \prod_{\varphi: F^{+} \rightarrow \mathbb{R}} X_{0}\right)(G,X)=(ResF+/Q⁡(G0)F+,∏φ:F+→RX0), where ( G 0 , X 0 ) G 0 , X 0 (G_(0),X_(0))\left(G_{0}, X_{0}\right)(G0,X0) is a Shimura datum and F + F + F^(+)F^{+}F+is a totally real field. As before, let p p ppp be a prime such that K p K p K_(p)K_{p}Kp is hyperspecial. In particular, p p ppp is unramified in F + F + F^(+)F^{+}F+. In addition, for simplicity we assume that G 0 , Q p G 0 , Q p G_(0,Q_(p))G_{0, \mathbb{Q}_{p}}G0,Qp is split (so for a place v v vvv of E E EEE above p , E v = Q p p , E v = Q p p,E_(v)=Q_(p)p, E_{v}=\mathbb{Q}_{p}p,Ev=Qp ). We let F F F\mathbb{F}F denote an algebraic closure of F p F p F_(p)\mathbb{F}_{p}Fp. Let { w i } i w i i {w_(i)}_(i)\left\{w_{i}\right\}_{i}{wi}i be the set of primes of F + F + F^(+)F^{+}F+above p p ppp, and let k i k i k_(i)k_{i}ki denote the residue field of w i w i w_(i)w_{i}wi. For each i i iii, we also fix an embedding ρ i : k i F ρ i : k i → F rho_(i):k_(i)rarrF\rho_{i}: k_{i} \rightarrow \mathbb{F}ρi:ki→F. Then there is a natural map
where S f i S f i S_(f_(i))\mathbb{S}_{f_{i}}Sfi is the permutation group on f i f i f_(i)f_{i}fi letters. Together with Theorem 2.3.3, one obtains the following result [64].
Theorem 3.2.2. There is an action of i ( Z f i S f i ) ∏ i   Z f i ⋊ S f i prod_(i)(Z^(f_(i))><|S_(f_(i)))\prod_{i}\left(\mathbb{Z}^{f_{i}} \rtimes \mathbb{S}_{f_{i}}\right)∏i(Zfi⋊Sfi) on H c ( S ¯ K , F ¯ , Z ) H c ∗ S ¯ K , F ¯ , Z â„“ H_(c)^(**)( bar(S)_(K, bar(F)),Z_(â„“))H_{c}^{*}\left(\overline{\mathscr{S}}_{K, \overline{\mathbb{F}}}, \mathbb{Z}_{\ell}\right)Hc∗(S¯K,F¯,Zâ„“) such that action of σ p σ p sigma_(p)\sigma_{p}σp factors as σ p = i σ p , i σ p = ∏ i   σ p , i sigma_(p)=prod_(i)sigma_(p,i)\sigma_{p}=\prod_{i} \sigma_{p, i}σp=∏iσp,i, where σ p , i = ( ( 1 , 0 , , 0 ) , ( 12 f i ) ) Z f i S f i σ p , i = ( 1 , 0 , … , 0 ) , 12 ⋯ f i ∈ Z f i ⋊ S f i sigma_(p,i)=((1,0,dots,0),(12 cdotsf_(i)))inZ^(f_(i))><|S_(f_(i))\sigma_{p, i}=\left((1,0, \ldots, 0),\left(12 \cdots f_{i}\right)\right) \in \mathbb{Z}^{f_{i}} \rtimes \mathbb{S}_{f_{i}}σp,i=((1,0,…,0),(12⋯fi))∈Zfi⋊Sfi. Each σ p , i f i σ p , i f i sigma_(p,i)^(f_(i))\sigma_{p, i}^{f_{i}}σp,ifi satisfies a polynomial equation similar to (3.2).
This theorem gives some shadow of the plectic cohomology conjecture of NekovářScholl [54].

3.3. Generic Tate cycles on mod p p p\boldsymbol{p}p fibers of Shimura varieties

In [64], we applied Theorem 2.3.3 to verify "generic" cases of Tate conjecture for the mod p mod p mod p\bmod pmodp fibers of many Shimura varieties. We use notations and (for simplicity) keep assumptions from Section 3.2. Let ( S ¯ K , k v ¯ ) pf S ¯ K , k v ¯ pf  ( bar(S)_(K, bar(k_(v))))^("pf ")\left(\overline{\mathscr{S}}_{K, \overline{k_{v}}}\right)^{\text {pf }}(S¯K,kv¯)pf  denote the perfection of S ¯ K , k v ¯ S ¯ K , k v ¯ bar(S)_(K, bar(k_(v)))\overline{\mathscr{S}}_{K, \overline{k_{v}}}S¯K,kv¯ (i.e., regard it as a perfect presheaf over A f f k f k ¯ v ) A f f k f k ¯ v {:Aff((kf)/( bar(k)_(v))))\left.\mathbf{A f f} \frac{\mathrm{kf}}{\bar{k}_{v}}\right)Affkfk¯v), then by attaching to every point of S ¯ K , k ¯ S ¯ K , k ¯ bar(S)_(K, bar(k))\overline{\mathscr{S}}_{K, \bar{k}}S¯K,k¯ an F F FFF-isocrystal with G G GGG-structure (see [ 37 , 64 ] [ 37 , 64 ] [37,64][37,64][37,64] ), one can define the so-called Newton map
N t : ( S ¯ K , k v ¯ ) p f B ( G Q p ) k v ¯ N t : S ¯ K , k v ¯ p f → B G Q p k v ¯ Nt:( bar(S)_(K, bar(k_(v))))^(pf)rarrB(G_(Q_(p)))_( bar(k_(v)))\mathrm{Nt}:\left(\overline{\mathscr{S}}_{K, \overline{k_{v}}}\right)^{\mathrm{pf}} \rightarrow \mathfrak{B}\left(G_{\mathbb{Q}_{p}}\right)_{\overline{k_{v}}}Nt:(S¯K,kv¯)pf→B(GQp)kv¯
Then the Newton stratification of B ( G Q p ) k v ¯ B G Q p k v ¯ B(G_(Q_(p)))_( bar(k_(v)))\mathfrak{B}\left(G_{\mathbb{Q}_{p}}\right)_{\overline{k_{v}}}B(GQp)kv¯ (see Section 2.1) induces a stratification of S ¯ K , k v ¯ S ¯ K , k v ¯ bar(S)_(K, bar(k_(v)))\overline{\mathscr{S}}_{K, \overline{k_{v}}}S¯K,kv¯ by locally closed subvarieties. It is known that the image of N t N t Nt\mathrm{Nt}Nt contains a unique basic element b b bbb and the corresponding subvarieties in S ¯ K , k v ¯ S ¯ K , k v ¯ bar(S)_(K, bar(k_(v)))\overline{\mathscr{S}}_{K, \overline{k_{v}}}S¯K,kv¯ is closed, called the basic Newton stratum, and denoted by S ¯ b S ¯ b bar(S)_(b)\overline{\mathscr{S}}_{b}S¯b.
Let m m mmm be the order of the action of the geometric Frobenius σ p σ p sigma_(p)\sigma_{p}σp on X ( T ^ ) X ∙ ( T ^ ) X∙( hat(T))\mathbb{X} \bullet(\hat{T})X∙(T^). Let
Λ p T a t e = { λ X ( T ^ ) i = 0 m 1 σ p i ( λ ) = 0 } X ( T ^ ) Λ p T a t e = λ ∈ X ∙ ( T ^ ) ∣ ∑ i = 0 m − 1   σ p i ( λ ) = 0 ⊂ X ∙ ( T ^ ) Lambda_(p)^(Tate)={lambda inX^(∙)(( hat(T)))∣sum_(i=0)^(m-1)sigma_(p)^(i)(lambda)=0}subX^(∙)( hat(T))\Lambda_{p}^{\mathrm{Tate}}=\left\{\lambda \in \mathbb{X}^{\bullet}(\hat{T}) \mid \sum_{i=0}^{m-1} \sigma_{p}^{i}(\lambda)=0\right\} \subset \mathbb{X}^{\bullet}(\hat{T})ΛpTate={λ∈X∙(T^)∣∑i=0m−1σpi(λ)=0}⊂X∙(T^)
For a representation V V VVV of G ^ Q G ^ Q â„“ hat(G)_(Q_(â„“))\hat{G}_{\mathbb{Q}_{\ell}}G^Qâ„“ and λ X ( T ^ ) λ ∈ X ∙ ( T ^ ) lambda inX∙( hat(T))\lambda \in \mathbb{X} \bullet(\hat{T})λ∈X∙(T^), let V ( λ ) V ( λ ) V(lambda)V(\lambda)V(λ) denote the λ λ lambda\lambdaλ-weight subspace of V V VVV (with respect to T ^ T ^ hat(T)\hat{T}T^ ), and let
V Tate = λ Λ p Tate V ( λ ) V Tate  = ⨁ λ ∈ Λ p Tate    V ( λ ) V^("Tate ")=bigoplus_(lambda inLambda_(p)^("Tate "))V(lambda)V^{\text {Tate }}=\bigoplus_{\lambda \in \Lambda_{p}^{\text {Tate }}} V(\lambda)VTate =⨁λ∈ΛpTate V(λ)
We are in particular interested in the condition V μ Tate 0 V μ Tate  ≠ 0 V_(mu)^("Tate ")!=0V_{\mu}^{\text {Tate }} \neq 0VμTate ≠0. As explained in the introduction of [64], under the conjectural Galois theoretic description of the cohomology of the
Shimura varieties (analogous to Conjecture 2.2.3), for a Hecke module π f Ï€ f pi_(f)\pi_{f}Ï€f whose Satake parameter at p p ppp is general enough, certain multiple a ( π f ) a Ï€ f a(pi_(f))a\left(\pi_{f}\right)a(Ï€f) of the dimension of this vector space should be equal to the dimension of the space of Tate classes in the π f Ï€ f pi_(f)\pi_{f}Ï€f-component of the middle dimensional compactly-supported cohomology of S ¯ K , k v ¯ S ¯ K , k v ¯ bar(S)_(K, bar(k_(v)))\overline{\mathscr{S}}_{K, \overline{k_{v}}}S¯K,kv¯. In addition, this space is usually large. For example, in the case G G GGG is an odd (projective) unitary group of signature ( i , n i ) ( i , n − i ) (i,n-i)(i, n-i)(i,n−i) over a quadratic imaginary field, the dimension of this space at an inert prime is ( n + 1 2 i ) ( n + 1 2 i ) (((n+1)/(2))/(i))\binom{\frac{n+1}{2}}{i}(n+12i).
For a (not necessarily irreducible) algebraic variety Z Z ZZZ of dimension d d ddd over an algebraically closed field, let H 2 d B M ( Z ) ( d ) H 2 d B M ( Z ) ( − d ) H_(2d)^(BM)(Z)(-d)H_{2 d}^{\mathrm{BM}}(Z)(-d)H2dBM(Z)(−d) denote the ( d ) ( − d ) (-d)(-d)(−d)-Tate twist of the top degree BorelMoore homology, which is the vector space spanned by the irreducible components of Z Z ZZZ. Now let X X XXX be a smooth variety of dimension d + r d + r d+rd+rd+r defined over a finite field k k kkk of q q qqq elements, and let Z X k ¯ Z ⊆ X k ¯ Z subeX_( bar(k))Z \subseteq X_{\bar{k}}Z⊆Xk¯ be a (not necessarily irreducible) projective subvariety of dimension d d ddd. There is the cycle class map
c l : H 2 d B M ( Z ) ( d ) j 1 H c 2 d ( X k ¯ , Q ( d ) ) σ q j =: T d ( X ) c l : H 2 d B M ( Z ) ( − d ) → ⋃ j ≥ 1   H c 2 d X k ¯ , Q â„“ ( d ) σ q j =: T â„“ d ( X ) cl:H_(2d)^(BM)(Z)(-d)rarruuu_(j >= 1)H_(c)^(2d)(X_( bar(k)),Q_(â„“)(d))^(sigma_(q)^(j))=:T_(â„“)^(d)(X)\mathrm{cl}: H_{2 d}^{\mathrm{BM}}(Z)(-d) \rightarrow \bigcup_{j \geq 1} H_{c}^{2 d}\left(X_{\bar{k}}, \mathbb{Q}_{\ell}(d)\right)^{\sigma_{q}^{j}}=: T_{\ell}^{d}(X)cl:H2dBM(Z)(−d)→⋃j≥1Hc2d(Xk¯,Qâ„“(d))σqj=:Tâ„“d(X)
Theorem 3.3.1. We write d μ = dim X = 2 d d μ = dim ⁡ X = 2 d d_(mu)=dim X=2dd_{\mu}=\operatorname{dim} X=2 ddμ=dim⁡X=2d and r = dim V μ Tate r = dim ⁡ V μ Tate  r=dim V_(mu)^("Tate ")r=\operatorname{dim} V_{\mu}^{\text {Tate }}r=dim⁡VμTate .
(1) The basic Newton stratum S ¯ b S ¯ b bar(S)_(b)\overline{\mathscr{S}}_{b}S¯b of S ¯ K , k ¯ v S ¯ K , k ¯ v bar(S)_(K, bar(k)_(v))\overline{\mathscr{S}}_{K, \bar{k}_{v}}S¯K,k¯v is pure of dimension d. In particular, d d ddd is always an integer. In addition, there is an H K , Q H K , Q â„“ H_(K,Q_(â„“))H_{K, \mathbb{Q}_{\ell}}HK,Qâ„“-equivariant isomorphism
H 2 d B M ( S ¯ b ) ( d ) C ( G ( Q ) G ( A f ) / K , Q ) r H 2 d B M S ¯ b ( − d ) ≅ C G ′ ( Q ) ∖ G ′ A f / K , Q â„“ ⊕ r H_(2d)^(BM)( bar(S)_(b))(-d)~=C(G^(')(Q)\\G^(')(A_(f))//K,Q_(â„“))^(o+r)H_{2 d}^{\mathrm{BM}}\left(\overline{\mathscr{S}}_{b}\right)(-d) \cong C\left(G^{\prime}(\mathbb{Q}) \backslash G^{\prime}\left(\mathbb{A}_{f}\right) / K, \mathbb{Q}_{\ell}\right)^{\oplus r}H2dBM(S¯b)(−d)≅C(G′(Q)∖G′(Af)/K,Qâ„“)⊕r
where G G ′ G^(')G^{\prime}G′ is the finitely trivialized inner form of G G GGG with G R G R ′ G_(R)^(')G_{\mathbb{R}}^{\prime}GR′ is compact.
(2) Let π f Ï€ f pi_(f)\pi_{f}Ï€f be an irreducible module of H K , Q ¯ H K , Q ¯ â„“ H_(K, bar(Q)_(â„“))H_{K, \overline{\mathbb{Q}}_{\ell}}HK,Q¯ℓ, and let
H 2 d B M ( S ¯ b ) [ π f ] = Hom H K , Q ¯ ( π f , H 2 d B M ( S ¯ b ) ( d ) Q ¯ ) π f H 2 d B M S ¯ b Ï€ f = Hom H K , Q ¯ â„“ ⁡ Ï€ f , H 2 d B M S ¯ b ( − d ) Q ¯ â„“ ⊗ Ï€ f H_(2d)^(BM)( bar(S)_(b))[pi_(f)]=Hom_(H_(K, bar(Q))^(â„“))(pi_(f),H_(2d)^(BM)( bar(S)_(b))(-d)_( bar(Q)_(â„“)))oxpi_(f)H_{2 d}^{\mathrm{BM}}\left(\overline{\mathscr{S}}_{b}\right)\left[\pi_{f}\right]=\operatorname{Hom}_{H_{K, \overline{\mathbb{Q}}}^{\ell}}\left(\pi_{f}, H_{2 d}^{\mathrm{BM}}\left(\overline{\mathscr{S}}_{b}\right)(-d)_{\overline{\mathbb{Q}}_{\ell}}\right) \otimes \pi_{f}H2dBM(S¯b)[Ï€f]=HomHK,Q¯ℓ⁡(Ï€f,H2dBM(S¯b)(−d)Q¯ℓ)⊗πf
be the π f Ï€ f pi_(f)\pi_{f}Ï€f-isotypical component. Then the cycle class map
c l : H 2 d B M ( S ¯ b ) ( d ) T d ( S ¯ K ) c l : H 2 d B M S ¯ b ( − d ) → T â„“ d S ¯ K cl:H_(2d)^(BM)( bar(S)_(b))(-d)rarrT_(â„“)^(d)( bar(S)_(K))\mathrm{cl}: H_{2 d}^{\mathrm{BM}}\left(\overline{\mathscr{S}}_{b}\right)(-d) \rightarrow T_{\ell}^{d}\left(\overline{\mathscr{S}}_{K}\right)cl:H2dBM(S¯b)(−d)→Tâ„“d(S¯K)
restricted to H 2 d B M ( S ¯ b ) [ π f ] H 2 d B M S ¯ b Ï€ f H_(2d)^(BM)( bar(S)_(b))[pi_(f)]H_{2 d}^{\mathrm{BM}}\left(\overline{\mathscr{S}}_{b}\right)\left[\pi_{f}\right]H2dBM(S¯b)[Ï€f] is injective if the Satake parameter of π f , p Ï€ f , p pi_(f,p)\pi_{f, p}Ï€f,p (the component of π f Ï€ f pi_(f)\pi_{f}Ï€f at p p ppp ) is V μ V μ V_(mu)V_{\mu}Vμ-general.
(3) Assume that S h K ( G ) S h K ( G ) Sh_(K)(G)\mathrm{Sh}_{K}(G)ShK(G) is (essentially) a quaternionic Shimura variety or a Kottwitz arithmetic variety. Then the π f Ï€ f pi_(f)\pi_{f}Ï€f-isotypical component of the cycle class map is surjective to T d ( S K ) ¯ [ π f ] T â„“ d S K ¯ Ï€ f T_(â„“)^(d) bar((S_(K)))[pi_(f)]T_{\ell}^{d} \overline{\left(\mathscr{S}_{K}\right)}\left[\pi_{f}\right]Tâ„“d(SK)¯[Ï€f] if the Satake parameter of π f , p Ï€ f , p pi_(f,p)\pi_{f, p}Ï€f,p is strongly V μ V μ V_(mu)V_{\mu}Vμ-general. In particular, the Tate conjecture holds for these π f Ï€ f pi_(f)\pi_{f}Ï€f.
Remark 3.3.2. (1) For a representation V V VVV of G ^ G ^ hat(G)\hat{G}G^, the definitions of " V V VVV-general" and "strongly V V VVV-general" Satake parameters can be found in [64, DEFINITION 1.4.2]. Regular semisimple elements in c G | d = ( p , σ p ) c G d = p , σ p ^(c)G|_(d=(p,sigma_(p)))\left.{ }^{c} G\right|_{d=\left(p, \sigma_{p}\right)}cG|d=(p,σp) are always V V VVV-general, but not the converse. See [64, remark 1.4.3].
(2) Some special cases of the theorem were originally proved in [ 33 , 60 ] [ 33 , 60 ] [33,60][33,60][33,60].
The proof of this theorem relies on several different ingredients. Via the RapoportZink uniformization of the basic locus of a Shimura variety, part (2) can be reduced a question about irreducible components of certain affine Deligne-Lusztig varieties, which was studied in [64, §3]. The most difficult is part (2), which we proved by calculating the intersection numbers among all d d ddd-dimensional cycles in S ¯ b S ¯ b bar(S)_(b)\overline{\mathscr{S}}_{b}S¯b. These numbers can be encoded in an r × r r × r r xx rr \times rr×r matrix with entries in H K p c l H K p c l H_(K_(p))^(cl)H_{K_{p}}^{\mathrm{cl}}HKpcl. In general, it seems hopeless to calculate this matrix directly and explicitly. However, this matrix can be understood as the composition of certain morphisms in Coh ( Loc c G , p u r ) Coh ⁡ Loc c G , p u r Coh(Loc_(cG,p)^(ur))\operatorname{Coh}\left(\operatorname{Loc}_{c G, p}^{\mathrm{ur}}\right)Coh⁡(LoccG,pur). Namely, first we realize G ( Q ) G ( A ) / K G ′ ( Q ) ∖ G ′ ( A ) / K G^(')(Q)\\G^(')(A)//KG^{\prime}(\mathbb{Q}) \backslash G^{\prime}(\mathbb{A}) / KG′(Q)∖G′(A)/K as a Shimura set with μ = 0 μ ′ = 0 mu^(')=0\mu^{\prime}=0μ′=0 its Shimura cocharacter. Then using Theorem 2.3.3 (and the Satake isomorphism (2.3)), this matrix can be calculated as
Then one needs to determine when this pairing is nondegenerate, which itself is an interesting question in representation theory, whose solution relies on the study of the Chevellay's restriction map for vector-valued functions. The determinant of this matrix was calculated in [65]. Finally, part (3) was proved by comparing two trace formulas, the Lefschetz trace formula for G G GGG and the Arthur-Selberg trace formula for G G ′ G^(')G^{\prime}G′.
Example 3.3.3. Let G = U ( 1 , 2 r ) G = U ( 1 , 2 r ) G=U(1,2r)G=\mathrm{U}(1,2 r)G=U(1,2r) be the unitary group 7 7 ^(7){ }^{7}7 of ( 2 r + 1 ) ( 2 r + 1 ) (2r+1)(2 r+1)(2r+1)-variables associated to an imaginary quadratic extension E / Q E / Q E//QE / \mathbb{Q}E/Q, whose signature is ( 1 , 2 r ) ( 1 , 2 r ) (1,2r)(1,2 r)(1,2r) at infinity. It is equipped with a standard Shimura datum, giving a Shimura variety (after fixing a level K G ( A f ) K ⊂ G A f K sub G(A_(f))K \subset G\left(\mathbb{A}_{f}\right)K⊂G(Af) ). In particular, if r = 1 r = 1 r=1r=1r=1, this is (essentially) the Picard modular surface. Let p p ppp be a prime inert in E E EEE such that K p K p K_(p)K_{p}Kp is hyperspecial. In this case S ¯ b S ¯ b bar(S)_(b)\overline{\mathscr{S}}_{b}S¯b is a union of certain Deligne-Lusztig varieties, parametrized by G ( Q ) G ( A f ) / K G ′ ( Q ) ∖ G ′ A f / K G^(')(Q)\\G^(')(A_(f))//KG^{\prime}(\mathbb{Q}) \backslash G^{\prime}\left(\mathbb{A}_{f}\right) / KG′(Q)∖G′(Af)/K, where G = U ( 0 , 2 r + 1 ) G ′ = U ( 0 , 2 r + 1 ) G^(')=U(0,2r+1)G^{\prime}=\mathrm{U}(0,2 r+1)G′=U(0,2r+1) that is isomorphic to G G GGG at all finite places. The intersection patterns of these cycles inside S ¯ b S ¯ b bar(S)_(b)\overline{\mathscr{S}}_{b}S¯b were (essentially) given in [61] but the intersection numbers between these cycles are much harder to compute. In fact, we do not know how to compute them directly for general r r rrr, except applying Theorem 2.3.3 to this case. (The case r = 1 r = 1 r=1r=1r=1 can be handled directly.)
We have G ^ = G L 2 r + 1 G ^ = G L 2 r + 1 hat(G)=GL_(2r+1)\hat{G}=\mathrm{GL}_{2 r+1}G^=GL2r+1 on which σ p σ p sigma_(p)\sigma_{p}σp acts as A J ( A T ) 1 J A ↦ J A T − 1 J A|->J(A^(T))^(-1)JA \mapsto J\left(A^{T}\right)^{-1} JA↦J(AT)−1J, where J J JJJ is the antidiagonal matrix with all entries along the antidiagonal being 1 . The representation V μ V μ V_(mu)V_{\mu}Vμ is the standard representation of G L 2 r + 1 G L 2 r + 1 GL_(2r+1)\mathrm{GL}_{2 r+1}GL2r+1. One checks that dim V μ Tate = 1 dim ⁡ V μ Tate  = 1 dim V_(mu)^("Tate ")=1\operatorname{dim} V_{\mu}^{\text {Tate }}=1dim⁡VμTate =1 (which is consistent with the above mentioned parameterization of irreducible components of S ¯ b S ¯ b bar(S)_(b)\overline{\mathscr{S}}_{b}S¯b by G ( Q ) G ( A f ) / K ) G ′ ( Q ) ∖ G ′ A f / K {:G^(')(Q)\\G^(')(A_(f))//K)\left.G^{\prime}(\mathbb{Q}) \backslash G^{\prime}\left(\mathbb{A}_{f}\right) / K\right)G′(Q)∖G′(Af)/K). We identify the weight lattice of G ^ G ^ hat(G)\hat{G}G^ as Z 2 r + 1 Z 2 r + 1 Z^(2r+1)\mathbb{Z}^{2 r+1}Z2r+1 as usual. Then Hom Coh ( Loc c u r u r ( O , V μ ~ ) Hom Coh ⁡ Loc c u r u r ⁡ O , V μ ~ Hom_(Coh(Loc_(c)^(ur):})^(ur)(O,( widetilde(V_(mu))))\operatorname{Hom}_{\operatorname{Coh}\left(\operatorname{Loc}_{c}^{\mathrm{ur}}\right.}^{\mathrm{ur}}\left(\mathcal{O}, \widetilde{V_{\mu}}\right)HomCoh⁡(Loccurur⁡(O,Vμ~) is a free rank one module over Hom Coh ( Loc C , p u r u r ( O , O ) = H K p c l Q Hom Coh ⁡ Loc C , p u r u r ⁡ ( O , O ) = H K p c l ⊗ Q â„“ Hom_(Coh(Loc_(C,p)^(ur):})^(ur)(O,O)=H_(K_(p))^(cl)oxQâ„“\operatorname{Hom}_{\operatorname{Coh}\left(\operatorname{Loc}_{C, p}^{\mathrm{ur}}\right.}^{\mathrm{ur}}(\mathcal{O}, \mathcal{O})=H_{K_{p}}^{\mathrm{cl}} \otimes \mathbb{Q} \ellHomCoh⁡(LocC,purur⁡(O,O)=HKpcl⊗Qâ„“. Then a generator a in a in  a_("in ")\mathbf{a}_{\text {in }}ain  induces an H K , Q H K , Q â„“ H_(K,Q_(â„“))H_{K, \mathbb{Q}_{\ell}}HK,Qâ„“-equivariant homomorphism
S ( a i n ) : C ( G ( Q ) G ( A f ) / K ) H c 2 r ( S ¯ K , k ¯ v , Q ( r ) ) S a i n : C G ′ ( Q ) ∖ G ′ A f / K → H c 2 r S ¯ K , k ¯ v , Q â„“ ( r ) S(a_(in)):C(G^(')(Q)\\G^(')(A_(f))//K)rarrH_(c)^(2r)( bar(S)_(K, bar(k)_(v)),Q_(â„“)(r))S\left(\mathbf{a}_{\mathrm{in}}\right): C\left(G^{\prime}(\mathbb{Q}) \backslash G^{\prime}\left(\mathbb{A}_{f}\right) / K\right) \rightarrow H_{c}^{2 r}\left(\overline{\mathscr{S}}_{K, \bar{k}_{v}}, \mathbb{Q}_{\ell}(r)\right)S(ain):C(G′(Q)∖G′(Af)/K)→Hc2r(S¯K,k¯v,Qâ„“(r))
This is not an adjoint group so the example is not consistent with our assumption. But it is more convenient for the discussion here. The computations are essentially the same.
realizing the cycle class map of S ¯ b S ¯ b bar(S)_(b)\overline{\mathscr{S}}_{b}S¯b (up to a multiple). The module Hom Coh ( Loc c G , p u r ( V μ ~ , O ) Hom Coh ⁡ ( Loc c G , p ⁡ u r V μ ~ , O Hom_(Coh(Loc)^(c_(G,p))^(ur)(( widetilde(V_(mu))),O)\operatorname{Hom}_{\operatorname{Coh}(\operatorname{Loc}}^{c_{G, p}}{ }^{\mathrm{ur}}\left(\widetilde{V_{\mu}}, \mathcal{O}\right)HomCoh⁡(LoccG,p⁡ur(Vμ~,O) is also free of rank one over H K p , Q H K p , Q â„“ H_(K_(p),Q_(â„“))H_{K_{p}, \mathbb{Q}_{\ell}}HKp,Qâ„“. For a chosen generator a out a out  a_("out ")\mathbf{a}_{\text {out }}aout , the composition
S ( a out ) S ( a i n ) = S ( a o u t a i n ) S a out  ∘ S a i n = S a o u t ∘ a i n S(a_("out "))@S(a_(in))=S(a_(out)@a_(in))S\left(\mathbf{a}_{\text {out }}\right) \circ S\left(\mathbf{a}_{\mathrm{in}}\right)=S\left(\mathbf{a}_{\mathrm{out}} \circ \mathbf{a}_{\mathrm{in}}\right)S(aout )∘S(ain)=S(aout∘ain)
calculates the intersection matrix of those cycles from the irreducible components of S ¯ b S ¯ b bar(S)_(b)\overline{\mathscr{S}}_{b}S¯b.
The element h := a out a in H K p , Q h := a out  ∘ a in  ∈ H K p , Q ℓ h:=a_("out ")@a_("in ")inH_(K_(p),Q_(ℓ))h:=\mathbf{a}_{\text {out }} \circ \mathbf{a}_{\text {in }} \in H_{K_{p}, \mathbb{Q}_{\ell}}h:=aout ∘ain ∈HKp,Qℓ was explicitly computed in [65, EXAMPLE 6.4.2] (up to obvious modification and also via the Satake isomorphism (1.4)). Namely,
(3.3) h = p r ( r + 1 ) i = 0 r ( 1 ) i ( 2 i + 1 ) p ( i r ) ( r + i + 1 ) j = 0 r i [ 2 r + 1 2 j r i j ] t = p T p , j (3.3) h = p r ( r + 1 ) ∑ i = 0 r   ( − 1 ) i ( 2 i + 1 ) p ( i − r ) ( r + i + 1 ) ∑ j = 0 r − i   2 r + 1 − 2 j r − i − j t = − p T p , j {:(3.3)h=p^(r(r+1))sum_(i=0)^(r)(-1)^(i)(2i+1)p^((i-r)(r+i+1))sum_(j=0)^(r-i)[[2r+1-2j],[r-i-j]]_(t=-p)T_(p,j):}h=p^{r(r+1)} \sum_{i=0}^{r}(-1)^{i}(2 i+1) p^{(i-r)(r+i+1)} \sum_{j=0}^{r-i}\left[\begin{array}{c} 2 r+1-2 j \tag{3.3}\\ r-i-j \end{array}\right]_{t=-p} T_{p, j}(3.3)h=pr(r+1)∑i=0r(−1)i(2i+1)p(i−r)(r+i+1)∑j=0r−i[2r+1−2jr−i−j]t=−pTp,j
Here, T p , j = 1 K p λ j ( p ) K p T p , j = 1 K p λ j ( p ) K p T_(p,j)=1_(K_(p)lambda_(j)(p)K_(p))T_{p, j}=1_{K_{p} \lambda_{j}(p) K_{p}}Tp,j=1Kpλj(p)Kp, with λ i = ( 1 i , 0 2 r 2 i + 1 , ( 1 ) i ) λ i = 1 i , 0 2 r − 2 i + 1 , ( − 1 ) i lambda_(i)=(1^(i),0^(2r-2i+1),(-1)^(i))\lambda_{i}=\left(1^{i}, 0^{2 r-2 i+1},(-1)^{i}\right)λi=(1i,02r−2i+1,(−1)i), and [ n m ] t n m t [[n],[m]]_(t)\left[\begin{array}{l}n \\ m\end{array}\right]_{t}[nm]t is the t t ttt-analogue of the binomial coefficient given by
[ 0 ] t = 1 , [ n ] t = t n 1 t 1 , [ n ] t ! = [ n ] t [ n 1 ] t [ 1 ] t , [ n m ] t = [ n ] t ! [ n m ] t ! [ m ] t ! [ 0 ] t = 1 , [ n ] t = t n − 1 t − 1 , [ n ] t ! = [ n ] t [ n − 1 ] t ⋯ [ 1 ] t , n m t = [ n ] t ! [ n − m ] t ! [ m ] t ! [0]_(t)=1,quad[n]_(t)=(t^(n)-1)/(t-1),quad[n]_(t)!=[n]_(t)[n-1]_(t)cdots[1]_(t),quad[[n],[m]]_(t)=([n]_(t)!)/([n-m]_(t)![m]_(t)!)[0]_{t}=1, \quad[n]_{t}=\frac{t^{n}-1}{t-1}, \quad[n]_{t}!=[n]_{t}[n-1]_{t} \cdots[1]_{t}, \quad\left[\begin{array}{c} n \\ m \end{array}\right]_{t}=\frac{[n]_{t}!}{[n-m]_{t}![m]_{t}!}[0]t=1,[n]t=tn−1t−1,[n]t!=[n]t[n−1]t⋯[1]t,[nm]t=[n]t![n−m]t![m]t!
In other words, the intersection matrix of cycles in S ¯ b S ¯ b bar(S)_(b)\overline{\mathscr{S}}_{b}S¯b in this case is calculated by the Hecke operator (3.3).
On interesting consequence is this computation is the following consequence on the intersection theory of the finite Deligne-Lusztig varieties, for which we do not know a direct proof. Let W W WWW be a ( 2 r + 1 ) ( 2 r + 1 ) (2r+1)(2 r+1)(2r+1)-dimensional nondegenerate hermitian space over F p 2 F p 2 F_(p^(2))\mathbb{F}_{p^{2}}Fp2. Consider the following r r rrr-dimensional Deligne-Lusztig variety
D L r := { H W of dimension r H ( H ( p ) ) } D L r := H ⊂ W  of dimension  r ∣ H ⊆ H ( p ) ⊥ DL_(r):={H sub W" of dimension "r∣H sube(H^((p)))^(_|_)}\mathrm{DL}_{r}:=\left\{H \subset W \text { of dimension } r \mid H \subseteq\left(H^{(p)}\right)^{\perp}\right\}DLr:={H⊂W of dimension r∣H⊆(H(p))⊥}
where H ( p ) H ( p ) H^((p))H^{(p)}H(p) the pullback of H H HHH along the Frobenius. Let H H H\mathscr{H}H denote the corresponding universal subbundle of rank r r rrr. Let E = H ( p ) ( ( H ( p ) ) / H ) E = H ( p ) ⊗ H ( p ) ⊥ / H E=H^((p))ox((H^((p)))^(_|_)//H)\mathcal{E}=\mathscr{H}^{(p)} \otimes\left(\left(\mathscr{H}^{(p)}\right)^{\perp} / \mathscr{H}\right)E=H(p)⊗((H(p))⊥/H). Then we have
(3.4) D L r c r ( E ) = i = 0 r ( 1 ) i ( 2 i + 1 ) p i 2 + i [ 2 r + 1 r i ] t = p (3.4) ∫ D L r   c r ( E ) = ∑ i = 0 r   ( − 1 ) i ( 2 i + 1 ) p i 2 + i 2 r + 1 r − i t = − p {:(3.4)int_(DL_(r))c_(r)(E)=sum_(i=0)^(r)(-1)^(i)(2i+1)p^(i^(2)+i)[[2r+1],[r-i]]_(t=-p):}\int_{\mathrm{DL}_{r}} c_{r}(\mathcal{E})=\sum_{i=0}^{r}(-1)^{i}(2 i+1) p^{i^{2}+i}\left[\begin{array}{c} 2 r+1 \tag{3.4}\\ r-i \end{array}\right]_{t=-p}(3.4)∫DLrcr(E)=∑i=0r(−1)i(2i+1)pi2+i[2r+1r−i]t=−p

3.4. The Beilinson-Bloch-Kato conjecture for Rankin-Selberg motives

Let M M MMM be a rational pure Chow motive of weight -1 over a number field F F FFF. The Beilinson-Bloch-Kato conjecture, which is a far reaching generalization of the Birch and Swinnerton-Dyer conjecture, predicts deep relations between certain algebraic, analytic, and cohomological invariants attached to M M MMM :
  • the rational Chow group C H ( M ) 0 C H ( M ) 0 CH(M)^(0)\mathrm{CH}(M)^{0}CH(M)0 of homologically trivial cycles of M M MMM;
  • the L L LLL-function L ( s , M ) L ( s , M ) L(s,M)L(s, M)L(s,M) of M M MMM;
  • the Bloch-Kato Selmer group H f 1 ( F , H ( M ) ) H f 1 F , H â„“ ( M ) H_(f)^(1)(F,H_(â„“)(M))H_{f}^{1}\left(F, H_{\ell}(M)\right)Hf1(F,Hâ„“(M)) of the â„“ â„“\ellâ„“-adic realization H ( M ) H â„“ ( M ) H_(â„“)(M)H_{\ell}(M)Hâ„“(M) of M M MMM.
The Beilinson-Bloch conjecture predicts an equality
dim Q C H ( M ) 0 = ord s = 0 L ( s , M ) dim Q ⁡ C H ( M ) 0 = ord s = 0 ⁡ L ( s , M ) dim_(Q)CH(M)^(0)=ord_(s=0)L(s,M)\operatorname{dim}_{\mathbb{Q}} \mathrm{CH}(M)^{0}=\operatorname{ord}_{s=0} L(s, M)dimQ⁡CH(M)0=ords=0⁡L(s,M)
between the dimension of C H ( M ) 0 C H ( M ) 0 CH(M)^(0)\mathrm{CH}(M)^{0}CH(M)0 and the vanishing order of the L L LLL-function at the central point, while the Bloch-Kato conjecture predicts
ord s = 0 L ( s , M ) = dim Q H f 1 ( F , H ( M ) ) ord s = 0 ⁡ L ( s , M ) = dim Q ℓ ⁡ H f 1 F , H ℓ ( M ) ord_(s=0)L(s,M)=dim_(Q_(ℓ))H_(f)^(1)(F,H_(ℓ)(M))\operatorname{ord}_{s=0} L(s, M)=\operatorname{dim}_{\mathbb{Q}_{\ell}} H_{f}^{1}\left(F, H_{\ell}(M)\right)ords=0⁡L(s,M)=dimQℓ⁡Hf1(F,Hℓ(M))
In addition, the so-called â„“ â„“\ellâ„“-adic Abel-Jacobi map
AJ : C H ( M ) 0 Q H f 1 ( F , H ( M ) ) AJ ℓ : C H ( M ) 0 ⊗ Q ℓ → H f 1 F , H ℓ ( M ) AJ_(ℓ):CH(M)^(0)oxQ_(ℓ)rarrH_(f)^(1)(F,H_(ℓ)(M))\operatorname{AJ}_{\ell}: \mathrm{CH}(M)^{0} \otimes \mathbb{Q}_{\ell} \rightarrow H_{f}^{1}\left(F, H_{\ell}(M)\right)AJℓ:CH(M)0⊗Qℓ→Hf1(F,Hℓ(M))
should be an isomorphism.
This conjecture seems to be completely out of reach at the moment. For example, for a general motive it is still widely open whether the L L LLL-function has a meromorphic continuation to the whole complex plane so that the vanishing order of L ( s , M ) L ( s , M ) L(s,M)L(s, M)L(s,M) at s = 0 s = 0 s=0s=0s=0 makes sense. (This would follow from the Galois-to-automorphic direction of the Langlands correspondence for number fields.) Despite this, there have been many works testing this conjecture in various special cases, mostly for motives M M MMM of small rank. In the work [49], we verify certain cases of the above conjecture for Rankin-Selberg motives, which consist of a sequence of motives of arbitrarily large rank.
We assume that F / F + F / F + F//F^(+)F / F^{+}F/F+is a (nontrivial) CM extension with F + F + F^(+)F^{+}F+totally real in the sequel.
Theorem 3.4.1. Let A 1 , A 2 A 1 , A 2 A_(1),A_(2)A_{1}, A_{2}A1,A2 be two elliptic curves over F + F + F^(+)F^{+}F+. Assume that
(1) End F ¯ A i = Z End F ¯ ⁡ A i = Z End_( bar(F))A_(i)=Z\operatorname{End}_{\bar{F}} A_{i}=\mathbb{Z}EndF¯⁡Ai=Z and Hom F ¯ ( A 1 , A 2 ) = 0 Hom F ¯ ⁡ A 1 , A 2 = 0 Hom_( bar(F))(A_(1),A_(2))=0\operatorname{Hom}_{\bar{F}}\left(A_{1}, A_{2}\right)=0HomF¯⁡(A1,A2)=0;
(2) Sym n 1 A 1 Sym n − 1 ⁡ A 1 Sym^(n-1)A_(1)\operatorname{Sym}^{n-1} A_{1}Symn−1⁡A1 and Sym n A 2 Sym n ⁡ A 2 Sym^(n)A_(2)\operatorname{Sym}^{n} A_{2}Symn⁡A2 are modular;
(3) F + Q F + ≠ Q F^(+)!=QF^{+} \neq \mathbb{Q}F+≠Q if n 3 n ≥ 3 n >= 3n \geq 3n≥3.
Under these assumption, if L ( n , Sym n 1 A 1 × Sym n A 2 ) 0 L n , Sym n − 1 ⁡ A 1 × Sym n ⁡ A 2 ≠ 0 L(n,Sym^(n-1)A_(1)xxSym^(n)A_(2))!=0L\left(n, \operatorname{Sym}^{n-1} A_{1} \times \operatorname{Sym}^{n} A_{2}\right) \neq 0L(n,Symn−1⁡A1×Symn⁡A2)≠0, then for almost â„“ â„“\ellâ„“,
dim Q H f 1 ( F , Sym n 1 V ( A 1 ) Sym n V ( A 2 ) ( 1 n ) ) = 0 dim Q ℓ ⁡ H f 1 F , Sym n − 1 ⁡ V ℓ A 1 ⊗ Sym n ⁡ V ℓ A 2 ( 1 − n ) = 0 dim_(Q_(ℓ))H_(f)^(1)(F,Sym^(n-1)V_(ℓ)(A_(1))oxSym^(n)V_(ℓ)(A_(2))(1-n))=0\operatorname{dim}_{\mathbb{Q}_{\ell}} H_{f}^{1}\left(F, \operatorname{Sym}^{n-1} V_{\ell}\left(A_{1}\right) \otimes \operatorname{Sym}^{n} V_{\ell}\left(A_{2}\right)(1-n)\right)=0dimQℓ⁡Hf1(F,Symn−1⁡Vℓ(A1)⊗Symn⁡Vℓ(A2)(1−n))=0
Here V ( A i ) V â„“ A i V_(â„“)(A_(i))V_{\ell}\left(A_{i}\right)Vâ„“(Ai) denotes the rational Tate module of A i A i A_(i)A_{i}Ai as usual.
This theorem is in fact a consequence of a more general result concerning BlochKato Selmer groups of Galois representations associated to certain Rankin-Selberg automorphic representations, which we now discuss.
Recall that for an irreducible regular algebraic conjugate self-dual cuspidal (RACSDC) automorphic representation Π Î  Pi\PiΠ of G L n ( A F ) G L n A F GL_(n)(A_(F))\mathrm{GL}_{n}\left(\mathbb{A}_{F}\right)GLn(AF), one can attach a compatible system of Galois representations ρ Π , λ : Γ F GL n ( E λ ) ρ Π , λ : Γ F → GL n ⁡ E λ rho_(Pi,lambda):Gamma_(F)rarrGL_(n)(E_(lambda))\rho_{\Pi, \lambda}: \Gamma_{F} \rightarrow \operatorname{GL}_{n}\left(E_{\lambda}\right)ρΠ,λ:ΓF→GLn⁡(Eλ), where E C E ⊂ C E subCE \subset \mathbb{C}E⊂C is a large enough number field and λ λ lambda\lambdaλ is a prime of E E EEE (see [16]). Such E E EEE is called a strong coefficient field of Π Î  Pi\PiΠ, which in the situation considered below can be taken as the number field generated by Hecke eigenvalues of Π Î  Pi\PiΠ.
Theorem 3.4.2. Suppose that F + Q F + ≠ Q F^(+)!=QF^{+} \neq \mathbb{Q}F+≠Q if n 3 n ≥ 3 n >= 3n \geq 3n≥3. Let Π n Π n Pi_(n)\Pi_{n}Πn (resp. Π n + 1 ) Π n + 1 {:Pi_(n+1))\left.\Pi_{n+1}\right)Πn+1) be an RACSDC automorphic representation of G L n ( A F ) G L n A F GL_(n)(A_(F))\mathrm{GL}_{n}\left(\mathbb{A}_{F}\right)GLn(AF) (resp. G L n + 1 ( A F ) G L n + 1 A F GL_(n+1)(A_(F))\mathrm{GL}_{n+1}\left(\mathbb{A}_{F}\right)GLn+1(AF) ) with trivial infinitesimal character. Let E C E ⊆ C E subeCE \subseteq \mathbb{C}E⊆C be a strong coefficient field for both Π n Π n Pi_(n)\Pi_{n}Πn and Π n + 1 Π n + 1 Pi_(n+1)\Pi_{n+1}Πn+1. Let λ λ lambda\lambdaλ be an admissible prime of E E EEE with respect to Π := Π 0 × Π 1 Π := Π 0 × Π 1 Pi:=Pi_(0)xxPi_(1)\Pi:=\Pi_{0} \times \Pi_{1}Π:=Π0×Π1. Let ρ Π , λ := ρ Π n , λ E λ ρ Π n + 1 , λ ρ Π , λ := ρ Π n , λ ⊗ E λ ρ Π n + 1 , λ rho_(Pi,lambda):=rho_(Pi_(n),lambda)ox_(E_(lambda))rho_(Pi_(n+1),lambda)\rho_{\Pi, \lambda}:=\rho_{\Pi_{n}, \lambda} \otimes_{E_{\lambda}} \rho_{\Pi_{n+1}, \lambda}ρΠ,λ:=ρΠn,λ⊗EλρΠn+1,λ.
(1) If the Rankin-Selberg L L LLL-value L ( 1 2 , Π ) 0 8 L 1 2 , Π ≠ 0 8 L((1)/(2),Pi)!=0^(8)L\left(\frac{1}{2}, \Pi\right) \neq 0^{8}L(12,Π)≠08, then H f 1 ( F , ρ Π , λ ( n ) ) = 0 H f 1 F , ρ Π , λ ( n ) = 0 H_(f)^(1)(F,rho_(Pi,lambda)(n))=0H_{f}^{1}\left(F, \rho_{\Pi, \lambda}(n)\right)=0Hf1(F,ρΠ,λ(n))=0.
(2) If certain element Δ λ H f 1 ( F , ρ Π , λ ( n ) ) Δ λ ∈ H f 1 F , ρ Π , λ ( n ) Delta_(lambda)inH_(f)^(1)(F,rho_(Pi,lambda)(n))\Delta_{\lambda} \in H_{f}^{1}\left(F, \rho_{\Pi, \lambda}(n)\right)Δλ∈Hf1(F,ρΠ,λ(n)) (to be explained below) is non-zero, then H f 1 ( F , ρ Π , λ ( n ) ) H f 1 F , ρ Π , λ ( n ) H_(f)^(1)(F,rho_(Pi,lambda)(n))H_{f}^{1}\left(F, \rho_{\Pi, \lambda}(n)\right)Hf1(F,ρΠ,λ(n)) is generated by Δ λ Δ λ Delta_(lambda)\Delta_{\lambda}Δλ as an E λ E λ E_(lambda)E_{\lambda}Eλ-vector space.
The notion of admissible primes appearing in the theorem consists of a long list of assumptions, some of which are rather technical. Essentially, it guarantees that the Galois

and the reduction mod λ λ lambda\lambdaλ representation is suitably large and contains certain particular elements. (This is also related to the notion of V V VVV-general from Theorem 3.3.1.) Fortunately, in some favorable situations, one can show that all but finitely many primes are admissible. For example, this is the case considered in Theorem 3.4.1. For another case in pure automorphic setting, see [ 49 [ 49 [49[49[49, тнм. 1.1.7].
The proof of the theorem uses several different ingredients. The initial step for case (1) is to translate the analytic condition L ( 1 2 , Π ) 0 L 1 2 , Π ≠ 0 L((1)/(2),Pi)!=0L\left(\frac{1}{2}, \Pi\right) \neq 0L(12,Π)≠0 into a more algebraic condition via the global Gan-Gross-Prasad (GGP) conjecture. Namely, the GGP conjecture predicts that in this case, there exist a pair of hermitian spaces ( V n , V n + 1 ) V n , V n + 1 (V_(n),V_(n+1))\left(V_{n}, V_{n+1}\right)(Vn,Vn+1) over F F FFF that are totally positive definite at ∞ oo\infty∞, where V n + 1 = V n F v V n + 1 = V n ⊕ F v V_(n+1)=V_(n)o+FvV_{n+1}=V_{n} \oplus F vVn+1=Vn⊕Fv with ( v , v ) = 1 ( v , v ) = 1 (v,v)=1(v, v)=1(v,v)=1, and a tempered cuspidal automorphic representation π = π n × π n + 1 Ï€ = Ï€ n × Ï€ n + 1 pi=pi_(n)xxpi_(n+1)\pi=\pi_{n} \times \pi_{n+1}Ï€=Ï€n×πn+1 of the product of unitary groups G = U ( V n ) × U ( V n + 1 ) G = U V n × U V n + 1 G=U(V_(n))xx U(V_(n+1))G=U\left(V_{n}\right) \times U\left(V_{n+1}\right)G=U(Vn)×U(Vn+1), such that the period integral
[ Δ H ] : C c ( Sh ( G ) , E ) [ π ] E Δ H : C c ∗ ( Sh ⁡ ( G ) , E ) [ Ï€ ] → E [Delta_(H)]:C_(c)^(**)(Sh(G),E)[pi]rarr E\left[\Delta_{H}\right]: C_{c}^{*}(\operatorname{Sh}(G), E)[\pi] \rightarrow E[ΔH]:Cc∗(Sh⁡(G),E)[Ï€]→E
does not vanish, where H := U ( V n ) H := U V n H:=U(V_(n))H:=U\left(V_{n}\right)H:=U(Vn) embeds into G G GGG diagonally, which induces an embedding Δ H : Sh ( H ) Sh ( G ) Δ H : Sh ⁡ ( H ) ↪ Sh ⁡ ( G ) Delta_(H):Sh(H)↪Sh(G)\Delta_{H}: \operatorname{Sh}(H) \hookrightarrow \operatorname{Sh}(G)ΔH:Sh⁡(H)↪Sh⁡(G) of corresponding Shimura varieties (in fact, Shimura sets) with appropriately chosen level structures (here and later we omit level structures from the notations). We denote by [ Δ H Δ H [Delta_(H):}\left[\Delta_{H}\right.[ΔH ] the homology class of Sh ( G ) Sh ⁡ ( G ) Sh(G)\operatorname{Sh}(G)Sh⁡(G) given by Sh ( H ) Sh ⁡ ( H ) Sh(H)\operatorname{Sh}(H)Sh⁡(H) and write C c ( Sh ( G ) , E ) [ π ] C c ∗ ( Sh ⁡ ( G ) , E ) [ Ï€ ] C_(c)^(**)(Sh(G),E)[pi]C_{c}^{*}(\operatorname{Sh}(G), E)[\pi]Cc∗(Sh⁡(G),E)[Ï€] for the π Ï€ pi\piÏ€-isotypical component of the cohomology (i.e., functions) of Sh ( G ) Sh ⁡ ( G ) Sh(G)\operatorname{Sh}(G)Sh⁡(G). This conjecture was first proved in [73] under some local restrictions which are too restrictive for arithmetic applications. Those restrictions are all lifted in our recent work through some new techniques in the study of trace formulae [8].
The strategy then is to construct, for every m 1 m ≥ 1 m >= 1m \geq 1m≥1, (infinitely many) cohomology classes { Θ m p } p H 1 ( F , ( ρ Π , λ / λ m ) ( 1 ) ) Θ m p p ⊂ H 1 F , ρ Π , λ / λ m ∗ ( 1 ) {Theta_(m)^(p)}_(p)subH^(1)(F,(rho_(Pi,lambda)//lambda^(m))^(**)(1))\left\{\Theta_{m}^{p}\right\}_{p} \subset H^{1}\left(F,\left(\rho_{\Pi, \lambda} / \lambda^{m}\right)^{*}(1)\right){Θmp}p⊂H1(F,(ρΠ,λ/λm)∗(1)), where p p ppp are appropriately chosen primes and ( ) ( 1 ) ( − ) ∗ ( 1 ) (-)^(**)(1)(-)^{*}(1)(−)∗(1) denotes the usual Pontryagin duality twisted by the cyclotomic character, such that the local Tate pairing at p p ppp between Θ m p Θ m p Theta_(m)^(p)\Theta_{m}^{p}Θmp and Selmer classes of the Galois representation ρ Π , λ / λ m ρ Π , λ / λ m rho_(Pi,lambda)//lambda^(m)\rho_{\Pi, \lambda} / \lambda^{m}ρΠ,λ/λm is related to the above period integral. Then one can use Kolyvagin type argument (amplified in [ 47 , 49 ] [ 47 , 49 ] [47,49][47,49][47,49] ), with { Θ m p } Θ m p {Theta_(m)^(p)}\left\{\Theta_{m}^{p}\right\}{Θmp} served as annihilators of the Selmer groups, to conclude.
The construction of Θ m p Θ m p Theta_(m)^(p)\Theta_{m}^{p}Θmp uses the diagonal embedding of Shimura varieties
Δ H : Sh ( H ) Sh ( G ) Δ H ′ : Sh ⁡ H ′ ↪ Sh ⁡ G ′ Delta_(H^(')):Sh(H^('))↪Sh(G^('))\Delta_{H^{\prime}}: \operatorname{Sh}\left(H^{\prime}\right) \hookrightarrow \operatorname{Sh}\left(G^{\prime}\right)ΔH′:Sh⁡(H′)↪Sh⁡(G′)
where H G H ′ ↪ G ′ H^(')↪G^(')H^{\prime} \hookrightarrow G^{\prime}H′↪G′ are prime-to- p p ppp trivialized (extended pure) inner forms of H G H ⊂ G H sub GH \subset GH⊂G (see Definition 2.3.1). These Shimura varieties have parahoric level structures at p p ppp, and using
8 Here we use the automorphic normalization of the L L LLL-function.
the theory of local models (Section 3.1) one can show that their integral models are polysemistable at p p ppp and compute the sheaf of nearby cycles on their mod p p ppp fibers. Using many ingredients, including the understanding of (integral) cohomology of Sh ( G ) Sh ⁡ G ′ Sh(G^('))\operatorname{Sh}\left(G^{\prime}\right)Sh⁡(G′) over F ¯ F ¯ bar(F)\bar{F}F¯, the computations from Example 3.3.3 (in particular, (3.3) and (3.4)), and the Taylor-Wiles patching method [48], one shows that ( ρ Π , λ / λ m ) ( 1 ) ρ Π , λ / λ m ∗ ( 1 ) (rho_(Pi,lambda)//lambda^(m))^(**)(1)\left(\rho_{\Pi, \lambda} / \lambda^{m}\right)^{*}(1)(ρΠ,λ/λm)∗(1) does appear in the cohomology of Sh ( G ) Sh ⁡ G ′ Sh(G^('))\operatorname{Sh}\left(G^{\prime}\right)Sh⁡(G′) (the so-called arithmetic level raising for Π Î  Pi\PiΠ ), and that the diagonal cycle Δ H Δ H ′ Delta_(H^('))\Delta_{H^{\prime}}ΔH′, when localized at ( ρ Π , λ / λ m ) ( 1 ) ρ Π , λ / λ m ∗ ( 1 ) (rho_(Pi,lambda)//lambda^(m))^(**)(1)\left(\rho_{\Pi, \lambda} / \lambda^{m}\right)^{*}(1)(ρΠ,λ/λm)∗(1), does give the desired class Θ m p Θ m p Theta_(m)^(p)\Theta_{m}^{p}Θmp. We shall mention that this is consistent with conjectures in Sections 2.1 and 2.3, as coherent sheaves on Loc G , p O E / λ m Loc G , p ⊗ O E / λ m Loc_(G,p)oxO_(E)//lambda^(m)\operatorname{Loc}_{G, p} \otimes \mathcal{O}_{E} / \lambda^{m}LocG,p⊗OE/λm corresponding to c c ccc-ind K p G ( Q p ) ( O E / λ m ) K p G Q p O E / λ m K_(p)^(G(Q_(p)))(O_(E)//lambda^(m))K_{p}^{G\left(\mathbb{Q}_{p}\right)}\left(\mathcal{O}_{E} / \lambda^{m}\right)KpG(Qp)(OE/λm) and c c ccc-ind K p G ( Q p ) ( O E / λ m ) K p ′ G ′ Q p O E / λ m K_(p)^(')G^(')(Q_(p))(O_(E)//lambda^(m))K_{p}^{\prime} G^{\prime}\left(\mathbb{Q}_{p}\right)\left(\mathcal{O}_{E} / \lambda^{m}\right)Kp′G′(Qp)(OE/λm) are expected to be related exactly in this way.
We could also explain the class Δ λ Δ λ Delta_(lambda)\Delta_{\lambda}Δλ appearing in case (2). Namely, in this case we start with an embedding of Shimura varieties Δ H : Sh ( H ) Sh ( G ) Δ H : Sh ⁡ ( H ) ↪ Sh ⁡ ( G ) Delta_(H):Sh(H)↪Sh(G)\Delta_{H}: \operatorname{Sh}(H) \hookrightarrow \operatorname{Sh}(G)ΔH:Sh⁡(H)↪Sh⁡(G), where G G GGG is a product of unitary groups such that Π Î  Pi\PiΠ descends to a tempered cuspidal automorphic representation π Ï€ pi\piÏ€ appearing in the middle dimensional cohomology of S h G S h G Sh_(G)\mathrm{Sh}_{G}ShG. Then the π Ï€ pi\piÏ€-isotypical component of the cycle Δ H Δ H Delta_(H)\Delta_{H}ΔH is homologous to zero, and we let Δ λ = AJ λ ( Δ H [ π ] ) Δ λ = AJ λ ⁡ Δ H [ Ï€ ] Delta_(lambda)=AJ_(lambda)(Delta_(H)[pi])\Delta_{\lambda}=\operatorname{AJ}_{\lambda}\left(\Delta_{H}[\pi]\right)Δλ=AJλ⁡(ΔH[Ï€]). The strategy to prove case (2) then is to reduce it to case (1) via some similar arguments as before.

ACKNOWLEDGMENTS

The author would like to thank all of his collaborators, without whom many works reported in this survey article would not be possible. Besides, he would also like to thank Edward Frenkel, Dennis Gaitsgory, Xuhua He, Michael Rapoport, Peter Scholze for teaching him and for discussions on various parts of the Langlands program over years.

FUNDING

The work is partially supported by NSF under agreement No. DMS-1902239 and a Simons fellowship.

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XINWEN ZHU

Department of Mathematics, California Institute of Technology, Pasadena, CA 91125, USA, xzhu@caltech.edu

4. ALGEBRAIC AND COMPLEX GEOMETRY

MARC LEVINE

ABSTRACT

We give a survey of the development of motivic cohomology, motivic categories, and some of their recent descendants.

MATHEMATICS SUBJECT CLASSIFICATION 2020

Primary 14F42; Secondary 14C15, 19D55, 19D45, 55P42

KEYWORDS

Motivic cohomology, motives, K K KKK-theory, algebraic cycles, motivic homotopy theory

1. INTRODUCTION

Motivic cohomology arose out of a marriage of Grothendieck's ideas about motives with a circle of conjectures about special values of zeta functions and L L LLL-functions. It has since taken on a very active life of its own, spawning a multitude of developments and applications. My intention in this survey is to present some of the history of motivic cohomology and the framework that supports it, its current state, and some thoughts about its future directions. I will say very little about the initial impetus given by the conjectures about zeta functions and L L LLL-functions, as there are many others who are much better qualified to tell that story. I will also say next to nothing about the many applications motivic cohomology has seen: I think this would be like writing about the applications of cohomology up to, say, 1950, and would certainly make this already lengthy survey completely unmanageable.
My basic premise is that motivic cohomology is supposed to be universal cohomology for algebro-geometric objects. As "universal" depends on the universe one happens to find oneself in, motivic cohomology is an ever-evolving construct. My plan is to give a path through some of the various universes that have given rise to motivic cohomologies, to describe the resulting motivic cohomologies and put them in a larger, usually categorical, framework. Our path will branch into several directions, reflecting the different aspects of algebraic and arithmetic geometry that have been touched by this theory. We begin with the conjectures of Beilinson and Lichtenbaum about motivic complexes that give rise to the universal Bloch-Ogus cohomology theory on smooth varieties over a field, and the candidate complexes constructed by Bloch and Suslin. We then take up Voevodsky's triangulated category of motives over a field and the embedding of the motivic complexes and motivic cohomology in this framework. The next developments moving further in this direction give us motivic homotopy categories that tell us about "generalized motivic cohomology" for a much wider class of schemes, analogous to the development of the stable homotopy category and generalized cohomology for spaces; this includes a number of candidate theories for motivic cohomology over a general base-scheme. We conclude with three variations on our theme:
  • Milnor-Witt motives and Milnor-Witt motivic cohomology, incorporating information about quadratic forms,
  • Motives with modulus, relaxing the usual condition of homotopy invariance with respect to the affine line, and
  • p p ppp-adic, étale motivic cohomology in mixed characteristic ( 0 , p ) ( 0 , p ) (0,p)(0, p)(0,p), with its connection to p p ppp-adic Hodge theory.
This last example does not yet, as far as I know, have a categorical framework, while one for a motivic cohomology with modulus is still in development.
There is already an extensive literature on the early development of motives and motivic cohomology. It was not my intention here to cover this part in detail, but I include a section on this topic to give a quick overview for the sake of background, and to isolate a few main ideas so the reader could see how they have influenced later developments.
I would like to thank all those who helped me prepare this survey, especially Tom Bachmann, Federico Binda, Dustin Clausen, Thomas Geisser, Wataru Kai, Akhil Mathew, Hiroyasu Miyazaki, Matthew Morrow, and Shuji Saito. In spite of their efforts, I feel certain that a number of errors have crept in, which are, of course, all my responsibility. I hope that the reader will be able to repair them and continue on.

2. BACKGROUND AND HISTORY

2.1. The conjectures of Beilinson and Lichtenbaum

Beilinson pointed out in his 1983 paper "Higher regulators and values of L L LLL-functions" [13] that the existence of Gillet's Chern character [53] from algebraic K K KKK-theory to an arbitrary Bloch-Ogus cohomology theory [30] with coefficients in a Q Q Q\mathbb{Q}Q-algebra implies that one can form the universal Bloch-Ogus cohomology H μ a ( , Q ( b ) ) H μ a ( − , Q ( b ) ) H_(mu)^(a)(-,Q(b))H_{\mu}^{a}(-, \mathbb{Q}(b))Hμa(−,Q(b)) with Q Q Q\mathbb{Q}Q-coefficients by decomposing algebraic K K KKK-theory into its weight spaces for the Adams operations ψ k ψ k psi_(k)\psi_{k}ψk. In terms of the indexing, one has
H a ( X , Q ( b ) ) := K 2 b a ( X ) ( b ) H a ( X , Q ( b ) ) := K 2 b − a ( X ) ( b ) H^(a)(X,Q(b)):=K_(2b-a)(X)^((b))H^{a}(X, \mathbb{Q}(b)):=K_{2 b-a}(X)^{(b)}Ha(X,Q(b)):=K2b−a(X)(b)
where K 2 b a ( X ) ( b ) K 2 b a ( X ) Q K 2 b − a ( X ) ( b ) ⊂ K 2 b − a ( X ) Q K_(2b-a)(X)^((b))subK_(2b-a)(X)_(Q)K_{2 b-a}(X)^{(b)} \subset K_{2 b-a}(X)_{\mathbb{Q}}K2b−a(X)(b)⊂K2b−a(X)Q is the weight b b bbb eigenspace for the Adams operations
K 2 b a ( X ) ( b ) := { x K 2 b a ( X ) Q ψ k ( x ) = k b x } K 2 b − a ( X ) ( b ) := x ∈ K 2 b − a ( X ) Q ∣ ψ k ( x ) = k b â‹… x K_(2b-a)(X)^((b)):={x inK_(2b-a)(X)_(Q)∣psi_(k)(x)=k^(b)*x}K_{2 b-a}(X)^{(b)}:=\left\{x \in K_{2 b-a}(X)_{\mathbb{Q}} \mid \psi_{k}(x)=k^{b} \cdot x\right\}K2b−a(X)(b):={x∈K2b−a(X)Q∣ψk(x)=kbâ‹…x}
This raised the question of finding the universal integral Bloch-Ogus cohomology theory. Let S c h k S c h k Sch_(k)\mathrm{Sch}_{k}Schk denote the category of separated finite-type k k kkk-schemes with full subcategory S m k S m k Sm_(k)\mathrm{Sm}_{k}Smk of smooth k k kkk-schemes. Beilinson [13] and Lichtenbaum [87] independently conjectured that this universal theory H μ a ( , Z ( b ) ) H μ a ( − , Z ( b ) ) H_(mu)^(a)(-,Z(b))H_{\mu}^{a}(-, \mathbb{Z}(b))Hμa(−,Z(b)) should arise as the hypercohomology of a complex of sheaves X Γ X ( b ) X ↦ Γ X ( b ) X|->Gamma_(X)(b)X \mapsto \Gamma_{X}(b)X↦ΓX(b) on S m k S m k Sm_(k)\mathrm{Sm}_{k}Smk (for the Zariski or étale topology)
H μ a ( X , Z ( b ) ) := H a ( X , Γ X ( b ) ) H μ a ( X , Z ( b ) ) := H a X , Γ X ( b ) H_(mu)^(a)(X,Z(b)):=H^(a)(X,Gamma_(X)(b))H_{\mu}^{a}(X, \mathbb{Z}(b)):=\mathbb{H}^{a}\left(X, \Gamma_{X}(b)\right)Hμa(X,Z(b)):=Ha(X,ΓX(b))
with the Γ X ( b ) Γ X ( b ) Gamma_(X)(b)\Gamma_{X}(b)ΓX(b) satisfying a number of axioms. We give Beilinson's list of axioms for motivic complexes in the Zariski topology (axiom iv ( p ) ( p ) (p)(p)(p) was added by Milne [90, $2]):
(i) In the derived category of sheaves on X , Γ ( 0 ) X , Γ ( 0 ) X,Gamma(0)X, \Gamma(0)X,Γ(0) is the constant sheaf Z Z Z\mathbb{Z}Z on S m k S m k Sm_(k)\mathrm{Sm}_{k}Smk, Γ ( 1 ) = G m [ 1 ] Γ ( 1 ) = G m [ − 1 ] Gamma(1)=G_(m)[-1]\Gamma(1)=\mathbb{G}_{m}[-1]Γ(1)=Gm[−1] and Γ ( n ) = 0 Γ ( n ) = 0 Gamma(n)=0\Gamma(n)=0Γ(n)=0 for n < 0 n < 0 n < 0n<0n<0.
(ii) The graded object Γ ( ) := [ X n 0 Γ X ( n ) ] Γ ( ∗ ) := X ↦ ⨁ n ≥ 0   Γ X ( n ) Gamma(**):=[X|->bigoplus_(n >= 0)Gamma_(X)(n)]\Gamma(*):=\left[X \mapsto \bigoplus_{n \geq 0} \Gamma_{X}(n)\right]Γ(∗):=[X↦⨁n≥0ΓX(n)] is a commutative graded ring in the derived category of sheaves on S m k S m k Sm_(k)\mathrm{Sm}_{k}Smk.
(iii) The cohomology sheaves H m ( Γ ( n ) ) H m ( Γ ( n ) ) H^(m)(Gamma(n))\mathscr{H}^{m}(\Gamma(n))Hm(Γ(n)) are zero for m > n m > n m > nm>nm>n and for m 0 m ≤ 0 m <= 0m \leq 0m≤0 if n > 0 n > 0 n > 0n>0n>0; H n ( Γ ( n ) ) H n ( Γ ( n ) ) H^(n)(Gamma(n))\mathscr{H}^{n}(\Gamma(n))Hn(Γ(n)) is the sheaf of Milnor K K KKK-groups X K n , X M X ↦ K n , X M X|->K_(n,X)^(M)X \mapsto \mathcal{K}_{n, X}^{M}X↦Kn,XM.
(iv)(a) Letting α : S m k , ét S m k , Z a r α : S m k ,  ét  → S m k , Z a r alpha:Sm_(k," ét ")rarrSm_(k,Zar)\alpha: \mathrm{Sm}_{k, \text { ét }} \rightarrow \mathrm{Sm}_{k, \mathrm{Zar}}α:Smk, ét →Smk,Zar be the change of topology morphism, the étale sheafification Γ ( n ) ét := α Γ ( n ) Γ ( n ) ét  := α ∗ Γ ( n ) Gamma(n)_("ét "):=alpha^(**)Gamma(n)\Gamma(n)_{\text {ét }}:=\alpha^{*} \Gamma(n)Γ(n)ét :=α∗Γ(n) of Γ ( n ) Γ ( n ) Gamma(n)\Gamma(n)Γ(n) satisfies Γ ( n ) ett / m μ m n Γ ( n ) ett  / m ≅ μ m ⊗ n Gamma(n)_("ett ")//m~=mu_(m)^(ox n)\Gamma(n)_{\text {ett }} / m \cong \mu_{m}^{\otimes n}Γ(n)ett /m≅μm⊗n for m m mmm prime to the characteristic, where μ m μ m mu_(m)\mu_{m}μm is the étale sheaf of m m mmm th roots of unity.
(iv)(b) For m m mmm prime to the characteristic, the natural map Γ ( n ) / m R α Γ ( n ) ett / m Γ ( n ) / m → R α ∗ Γ ( n ) ett  / m Gamma(n)//m rarr Ralpha_(**)Gamma(n)_("ett ")//m\Gamma(n) / m \rightarrow R \alpha_{*} \Gamma(n)_{\text {ett }} / mΓ(n)/m→Rα∗Γ(n)ett /m induces an isomorphism Γ ( n ) / m τ n R α Γ ( n ) et / m Γ ( n ) / m → Ï„ ≤ n R α ∗ Γ ( n ) et  / m Gamma(n)//m rarrtau_( <= n)Ralpha_(**)Gamma(n)_("et ")//m\Gamma(n) / m \rightarrow \tau_{\leq n} R \alpha_{*} \Gamma(n)_{\text {et }} / mΓ(n)/m→τ≤nRα∗Γ(n)et /m. Integrally,
Z ( n ) R α Γ ( n ) ét Z ( n ) → R α ∗ Γ ( n ) ét  Z(n)rarr Ralpha_(**)Gamma(n)_("ét ")\mathbb{Z}(n) \rightarrow R \alpha_{*} \Gamma(n)_{\text {ét }}Z(n)→Rα∗Γ(n)ét  induces an isomorphism Γ ( n ) τ n R α Γ ( n ) ét Γ ( n ) → Ï„ ≤ n R α ∗ Γ ( n ) ét  Gamma(n)rarrtau_( <= n)Ralpha_(**)Gamma(n)_("ét ")\Gamma(n) \rightarrow \tau_{\leq n} R \alpha_{*} \Gamma(n)_{\text {ét }}Γ(n)→τ≤nRα∗Γ(n)ét  and
R n + 1 α Γ ( n ) ét = 0 R n + 1 α ∗ Γ ( n ) ét  = 0 R^(n+1)alpha_(**)Gamma(n)_("ét ")=0R^{n+1} \alpha_{*} \Gamma(n)_{\text {ét }}=0Rn+1α∗Γ(n)ét =0
(iv)( p p ppp ) For k k kkk of characteristic p > 0 p > 0 p > 0p>0p>0, let W v Ω log n W v Ω log  n W_(v)Omega_("log ")^(n)W_{v} \Omega_{\text {log }}^{n}WvΩlog n denote the ν ν nu\nuν-truncated logarithmic de Rham-Witt sheaf. The d log d log d logd \logdlog map d log : K n M / p n W ν Ω log n d log : K n M / p n → W ν Ω log n d log:K_(n)^(M)//p^(n)rarrW_(nu)Omega_(log)^(n)d \log : \mathcal{K}_{n}^{M} / p^{n} \rightarrow W_{\nu} \Omega_{\log }^{n}dlog:KnM/pn→WνΩlogn induces via (ii) a map Γ ( n ) / p ν W v Ω log n [ n ] Γ ( n ) / p ν → W v Ω log n [ − n ] Gamma(n)//p^(nu)rarrW_(v)Omega_(log)^(n)[-n]\Gamma(n) / p^{\nu} \rightarrow W_{v} \Omega_{\log }^{n}[-n]Γ(n)/pν→WvΩlogn[−n], which is an isomorphism.
One then defines motivic cohomology by
H p ( X , Z ( q ) ) := H p ( X Z a r , Γ X ( q ) ) H p ( X , Z ( q ) ) := H p X Z a r , Γ X ( q ) H^(p)(X,Z(q)):=H^(p)(X_(Zar),Gamma_(X)(q))H^{p}(X, \mathbb{Z}(q)):=\mathbb{H}^{p}\left(X_{\mathrm{Zar}}, \Gamma_{X}(q)\right)Hp(X,Z(q)):=Hp(XZar,ΓX(q))
(v) There should also be a spectral sequence starting with integral motivic cohomology and converging to algebraic K K KKK-theory, analogous to the Atiyah-Hirzebruch spectral sequence from singular cohomology to topological K K KKK-theory. Explicitly, this should be
E 2 p , q := H p q ( X , Z ( q ) ) K p q ( X ) E 2 p , q := H p − q ( X , Z ( − q ) ) ⇒ K − p − q ( X ) E_(2)^(p,q):=H^(p-q)(X,Z(-q))=>K_(-p-q)(X)E_{2}^{p, q}:=H^{p-q}(X, \mathbb{Z}(-q)) \Rightarrow K_{-p-q}(X)E2p,q:=Hp−q(X,Z(−q))⇒K−p−q(X)
This spectral sequence should degenerate rationally, and give an isomorphism
H p ( X , Q ( q ) ) := H p ( X , Z ( q ) ) Z Q K 2 q p ( X ) ( q ) H p ( X , Q ( q ) ) := H p ( X , Z ( q ) ) ⊗ Z Q ≅ K 2 q − p ( X ) ( q ) H^(p)(X,Q(q)):=H^(p)(X,Z(q))ox_(Z)Q~=K_(2q-p)(X)^((q))H^{p}(X, \mathbb{Q}(q)):=H^{p}(X, \mathbb{Z}(q)) \otimes_{\mathbb{Z}} \mathbb{Q} \cong K_{2 q-p}(X)^{(q)}Hp(X,Q(q)):=Hp(X,Z(q))⊗ZQ≅K2q−p(X)(q)
The vanishing H m ( Γ ( n ) ) = 0 H m ( Γ ( n ) ) = 0 H^(m)(Gamma(n))=0\mathscr{H}^{m}(\Gamma(n))=0Hm(Γ(n))=0 for n > 0 n > 0 n > 0n>0n>0 and m 0 m ≤ 0 m <= 0m \leq 0m≤0 is the integral Beilinson-Soulé vanishing conjecture. The mod m m mmm-part of axiom (iv)(b) is known as the Beilinson-Lichtenbaum conjecture; this implies the integral part of (iv)(b) with the exception of the vanishing of R n + 1 α Γ ( n ) ét R n + 1 α ∗ Γ ( n ) ét  R^(n+1)alpha_(**)Gamma(n)_("ét ")R^{n+1} \alpha_{*} \Gamma(n)_{\text {ét }}Rn+1α∗Γ(n)ét , which is known as Hilbert's theorem 90 for the motivic complexes. In weight n = 1 n = 1 n=1n=1n=1, with the identity Γ ( 1 ) = G m [ 1 ] Γ ( 1 ) = G m [ − 1 ] Gamma(1)=G_(m)[-1]\Gamma(1)=\mathbb{G}_{m}[-1]Γ(1)=Gm[−1], this translates into the classical Hilbert theorem 90
H e t 1 ( O , G m ) = 0 H e t 1 O , G m = 0 H_(et)^(1)(O,G_(m))=0H_{\mathrm{et}}^{1}\left(\mathcal{O}, \mathbb{G}_{m}\right)=0Het1(O,Gm)=0
for O O O\mathcal{O}O a local ring, while the mod m m mmm part of (iv)(b) follows from the Kummer sequence of étale sheaves
1 μ m G m × m G m 1 1 → μ m → G m → × m G m → 1 1rarrmu_(m)rarrG_(m)rarr"xx m"G_(m)rarr11 \rightarrow \mu_{m} \rightarrow \mathbb{G}_{m} \xrightarrow{\times m} \mathbb{G}_{m} \rightarrow 11→μm→Gm→×mGm→1
In light of axiom (iii), the Merkurjev-Suslin theorem [89, Ñ‚HEOREM 14.1] settled the degree 2 ≥ 2 >= 2\geq 2≥2 part of (iv)(b) for n = 2 n = 2 n=2n=2n=2 even before the complex Γ Î“ Gamma\GammaΓ (2) was defined.
Beilinson [14, $5.10] rephrased and refined these conjectures to a categorical statement, invoking a conjectural category of mixed motivic sheaves, and an embedding of the hypercohomology of the Beilinson-Lichtenbaum complexes into a categorical framework.
In this framework, motivic cohomology should arise via an abelian tensor category of motivic sheaves on Sch S , X Sh mot ( X ) Sch S , X ↦ Sh mot  ⁡ ( X ) Sch_(S),X|->Sh^("mot ")(X)\operatorname{Sch}_{S}, X \mapsto \operatorname{Sh}^{\text {mot }}(X)SchS,X↦Shmot ⁡(X), admitting the six functor formalism of Grothendieck, f , f , f ! , f ! , H o m , f ∗ , f ∗ , f ! , f ! , H o m , ⊗ f^(**),f_(**),f_(!),f^(!),Hom,oxf^{*}, f_{*}, f_{!}, f^{!}, \mathscr{H o m}, \otimesf∗,f∗,f!,f!,Hom,⊗, on the derived categories. There should be Tate objects Z X ( n ) Sh mot ( X ) Z X ( n ) ∈ Sh mot  ⁡ ( X ) Z_(X)(n)inSh^("mot ")(X)\mathbb{Z}_{X}(n) \in \operatorname{Sh}^{\text {mot }}(X)ZX(n)∈Shmot ⁡(X), and objects M ( X ) := p X ! p X ! Z S ( 0 ) M ( X ) := p X ! p X ! Z S ( 0 ) M(X):=p_(X!)p_(X)^(!)Z_(S)(0)M(X):=p_{X!} p_{X}^{!} \mathbb{Z}_{S}(0)M(X):=pX!pX!ZS(0) in the derived category of Sh m o t ( S ) , p X : X S Sh m o t ⁡ ( S ) , p X : X → S Sh^(mot)(S),p_(X):X rarr S\operatorname{Sh}^{\mathrm{mot}}(S), p_{X}: X \rightarrow SShmot⁡(S),pX:X→S the structure morphism, and motivic cohomology should arise as the Hom-groups
H μ a ( X , Z ( b ) ) = Hom D ( S h m o t ( S ) ) ( M ( X ) , Z S ( b ) [ a ] ) H μ a ( X , Z ( b ) ) = Hom D S h m o t ( S ) ⁡ M ( X ) , Z S ( b ) [ a ] H_(mu)^(a)(X,Z(b))=Hom_(D(Sh^(mot)(S)))(M(X),Z_(S)(b)[a])H_{\mu}^{a}(X, \mathbb{Z}(b))=\operatorname{Hom}_{D\left(\mathrm{Sh}^{\mathrm{mot}}(S)\right)}\left(M(X), \mathbb{Z}_{S}(b)[a]\right)Hμa(X,Z(b))=HomD(Shmot(S))⁡(M(X),ZS(b)[a])
For X X XXX smooth over S S SSS, this gives the identity
H μ a ( X , Z ( b ) ) = Ext S h mot ( X ) a ( Z X ( 0 ) , Z X ( b ) ) H μ a ( X , Z ( b ) ) = Ext S h mot ( X ) a ⁡ Z X ( 0 ) , Z X ( b ) H_(mu)^(a)(X,Z(b))=Ext_(Sh^(mot)(X))^(a)(Z_(X)(0),Z_(X)(b))H_{\mu}^{a}(X, \mathbb{Z}(b))=\operatorname{Ext}_{\mathrm{Sh}^{\operatorname{mot}}(X)}^{a}\left(\mathbb{Z}_{X}(0), \mathbb{Z}_{X}(b)\right)Hμa(X,Z(b))=ExtShmot(X)a⁡(ZX(0),ZX(b))
This is a very strong statement, with implications that have not been verified to this day. For instance, the vanishing of Ext A a ( , ) Ext A a ⁡ ( − , − ) Ext_(A)^(a)(-,-)\operatorname{Ext}_{\mathcal{A}}^{a}(-,-)ExtAa⁡(−,−) for an abelian category A A A\mathcal{A}A and for a < 0 a < 0 a < 0a<0a<0 gives a vanishing H m ( Z S ( n ) ) = 0 H m Z S ( n ) = 0 H^(m)(Z_(S)(n))=0\mathscr{H}^{m}\left(\mathbb{Z}_{S}(n)\right)=0Hm(ZS(n))=0 for m < 0 m < 0 m < 0m<0m<0. The stronger vanishing posited by axiom (iii) above (with Q Q Q\mathbb{Q}Q-coefficients) is the Beilinson-Soulé vanishing conjecture, and even the weak version is only known for weight n = 1 n = 1 n=1n=1n=1 (for which the strong version holds).
Beilinson's conjecture on categories of motivic sheaves is still an open problem. However, other than the integral Beilinson-Soulé vanishing conjecture, the axioms do not rely on the existence of an abelian category of motivic sheaves, and can be framed in the setting of a functorial assignment X D M ( X ) X ↦ D M ( X ) X|->DM(X)X \mapsto \mathrm{DM}(X)X↦DM(X) from S S SSS-schemes to tensor-triangulated categories. Such a functor has been constructed and the axioms (except for the vanishing conjectures) have been verified. We will discuss this construction in Section 2.4.

2.2. Bloch's higher Chow groups and Suslin homology

The first good definition of motivic cohomology complexes was given by Spencer Bloch, in his landmark 1985 paper "Algebraic cycles and higher Chow groups" [24]. As

cycles modulo rational equivalence.
For X X XXX a finite type k k kkk-scheme, recall that the group of dimension d d ddd algebraic cycles on X , Z d ( X ) X , Z d ( X ) X,Z_(d)(X)X, \mathrm{Z}_{d}(X)X,Zd(X), is the free abelian group on the integral closed subschemes Z Z ZZZ of X X XXX of dimension d d ddd over k k kkk. The group of cycles modulo rational equivalence, C H d ( X ) C H d ( X ) CH_(d)(X)\mathrm{CH}_{d}(X)CHd(X), has the following presentation. Let n Δ n n ↦ Δ n n|->Delta^(n)n \mapsto \Delta^{n}n↦Δn be the cosimplicial scheme of algebraic n n nnn-simplices
Δ n := Spec Z [ t 0 , , t n ] / i = 0 n t i 1 A Z n Δ n := Spec ⁡ Z t 0 , … , t n / ∑ i = 0 n   t i − 1 ≅ A Z n Delta^(n):=Spec Z[t_(0),dots,t_(n)]//sum_(i=0)^(n)t_(i)-1~=A_(Z)^(n)\Delta^{n}:=\operatorname{Spec} \mathbb{Z}\left[t_{0}, \ldots, t_{n}\right] / \sum_{i=0}^{n} t_{i}-1 \cong \mathbb{A}_{\mathbb{Z}}^{n}Δn:=Spec⁡Z[t0,…,tn]/∑i=0nti−1≅AZn
The coface and codegeneracy maps are defined just as for the usual real simplices Δ top n R n Δ top  n ⊂ R n Delta_("top ")^(n)subR^(n)\Delta_{\text {top }}^{n} \subset \mathbb{R}^{n}Δtop n⊂Rn. A face of Δ n Δ n Delta^(n)\Delta^{n}Δn is a closed subscheme defined by the vanishing of some of the t i t i t_(i)t_{i}ti. Let z d ( X , n ) z d ( X , n ) z_(d)(X,n)z_{d}(X, n)zd(X,n) be the subgroup of the ( n + d ) ( n + d ) (n+d)(n+d)(n+d)-dimensional algebraic cycles Z n + d ( X × Δ n ) Z n + d X × Δ n Z_(n+d)(X xxDelta^(n))Z_{n+d}\left(X \times \Delta^{n}\right)Zn+d(X×Δn) generated by the integral closed W X × Δ n W ⊂ X × Δ n W sub X xxDelta^(n)W \subset X \times \Delta^{n}W⊂X×Δn such that dim W X × F = m + d dim ⁡ W ∩ X × F = m + d dim W nn X xx F=m+d\operatorname{dim} W \cap X \times F=m+ddim⁡W∩X×F=m+d for each m m mmm-dimensional face F F FFF (or the intersection is empty). For cycles w z d ( X , n ) w ∈ z d ( X , n ) w inz_(d)(X,n)w \in z_{d}(X, n)w∈zd(X,n), the face condition gives a well-defined pullback ( Id X × g ) : z d ( X , n ) z d ( X , m ) Id X × g ∗ : z d ( X , n ) → z d ( X , m ) (Id_(X)xx g)^(**):z_(d)(X,n)rarrz_(d)(X,m)\left(\operatorname{Id}_{X} \times g\right)^{*}: z_{d}(X, n) \rightarrow z_{d}(X, m)(IdX×g)∗:zd(X,n)→zd(X,m) for each map g : Δ m Δ n g : Δ m → Δ n g:Delta^(m)rarrDelta^(n)g: \Delta^{m} \rightarrow \Delta^{n}g:Δm→Δn in the cosimplicial structure, forming the simplicial abelian group n z d ( X , n ) n ↦ z d ( X , n ) n|->z_(d)(X,n)n \mapsto z_{d}(X, n)n↦zd(X,n) and giving the associated chain complex z d ( X , ) z d ( X , ∗ ) z_(d)(X,**)z_{d}(X, *)zd(X,∗), Bloch's cycle complex. The degree 0 and 1 terms of z d ( X , ) z d ( X , ∗ ) z_(d)(X,**)z_{d}(X, *)zd(X,∗) give our promised presentation of C H d ( X ) C H d ( X ) CH_(d)(X)\mathrm{CH}_{d}(X)CHd(X),
H 0 ( z d ( X , ) ) = C H d ( X ) H 0 z d ( X , ∗ ) = C H d ( X ) H_(0)(z_(d)(X,**))=CH_(d)(X)H_{0}\left(z_{d}(X, *)\right)=\mathrm{CH}_{d}(X)H0(zd(X,∗))=CHd(X)
and Bloch defines his higher Chow group C H d ( X , n ) C H d ( X , n ) CH_(d)(X,n)\mathrm{CH}_{d}(X, n)CHd(X,n) as
C H d ( X , n ) := H n ( z d ( X , ) ) C H d ( X , n ) := H n z d ( X , ∗ ) CH_(d)(X,n):=H_(n)(z_(d)(X,**))\mathrm{CH}_{d}(X, n):=H_{n}\left(z_{d}(X, *)\right)CHd(X,n):=Hn(zd(X,∗))
If X X XXX has pure dimension N N NNN over k k kkk, we index by codimension
z q ( X , ) := z N q ( X , ) ; C H q ( X , n ) := C H N q ( X , n ) z q ( X , ∗ ) := z N − q ( X , ∗ ) ; C H q ( X , n ) := C H N − q ( X , n ) z^(q)(X,**):=z_(N-q)(X,**);quadCH^(q)(X,n):=CH_(N-q)(X,n)z^{q}(X, *):=z_{N-q}(X, *) ; \quad \mathrm{CH}^{q}(X, n):=\mathrm{CH}_{N-q}(X, n)zq(X,∗):=zN−q(X,∗);CHq(X,n):=CHN−q(X,n)
With some technical difficulties due to the necessity of invoking moving lemmas to allow for pullback morphisms, the assignment
X z q ( X , 2 q ) X ↦ z q ( X , 2 q − ∗ ) X|->z^(q)(X,2q-**)X \mapsto z^{q}(X, 2 q-*)X↦zq(X,2q−∗)
can be modified via isomorphisms in the derived category to a presheaf of cohomological complexes Z B l ( q ) Z B l ( q ) Z_(Bl)(q)\mathbb{Z}_{B l}(q)ZBl(q) on S m k S m k Sm_(k)\mathrm{Sm}_{k}Smk.
Following a long series of works [ 25 , 29 , 43 , 52 , 94 , 96 , 112 , 113 , 115 117 , 120 , 121 , 123 125 [ 25 , 29 , 43 , 52 , 94 , 96 , 112 , 113 , 115 − 117 , 120 , 121 , 123 − 125 [25,29,43,52,94,96,112,113,115-117,120,121,123-125[25,29,43,52,94,96,112,113,115-117,120,121,123-125[25,29,43,52,94,96,112,113,115−117,120,121,123−125, 127] (see also [56, 102] for detailed discussions of the Bloch-Kato conjecture, the essential point in axiom (iv)(b) and the most difficult of the Beilinson axioms to prove), it has been shown that the complexes Z B I ( q ) Z B I ( q ) Z_(BI)(q)\mathbb{Z}_{\mathrm{BI}}(q)ZBI(q) satisfy all the Beilinson-Lichtenbaum-Milne axioms, except for the Beilinson-Soulé vanishing conjecture in axiom (iii).
After Bloch introduced his cycle complexes, Suslin [111] constructed an algebraic version of singular homology. For a k k kkk-scheme X X XXX, instead of a naive generalization of the singular chain complex of a topological space by taking the free abelian group on the morphisms Δ k n X Δ k n → X Delta_(k)^(n)rarr X\Delta_{k}^{n} \rightarrow XΔkn→X, Suslin's insight was to enlarge this to the abelian group of finite correspondences.
A subvariety W W WWW of a product Y × X Y × X Y xx XY \times XY×X of varieties (with Y Y YYY smooth) defines an irreducible finite correspondence from Y Y YYY to X X XXX if p 1 : W Y p 1 : W → Y p_(1):W rarr Yp_{1}: W \rightarrow Yp1:W→Y is finite and surjective to some irreducible component of Y Y YYY. The association y p 2 ( p 1 1 ( y ) ) y ↦ p 2 p 1 − 1 ( y ) y|->p_(2)(p_(1)^(-1)(y))y \mapsto p_{2}\left(p_{1}^{-1}(y)\right)y↦p2(p1−1(y)) can be thought of as a multivalued map from Y Y YYY to X X XXX.
The group of finite correspondences Cor k ( Y , X ) Cor k ⁡ ( Y , X ) Cor_(k)(Y,X)\operatorname{Cor}_{k}(Y, X)Cork⁡(Y,X) is defined as the free abelian group on the irreducible finite correspondences. Given a morphism f : Y Y f : Y ′ → Y f:Y^(')rarr Yf: Y^{\prime} \rightarrow Yf:Y′→Y, there is a pullback map f : Cor k ( Y , X ) Cor k ( Y , X ) f ∗ : Cor k ⁡ ( Y , X ) → Cor k ⁡ Y ′ , X f^(**):Cor_(k)(Y,X)rarrCor_(k)(Y^('),X)f^{*}: \operatorname{Cor}_{k}(Y, X) \rightarrow \operatorname{Cor}_{k}\left(Y^{\prime}, X\right)f∗:Cork⁡(Y,X)→Cork⁡(Y′,X), compatible with the interpretation as multivalued functions, and making Cor k ( , X ) Cor k ⁡ ( − , X ) Cor_(k)(-,X)\operatorname{Cor}_{k}(-, X)Cork⁡(−,X) into a contravariant functor from smooth varieties over k k kkk to abelian groups.
Suslin defines C n Sus ( X ) := Cor k ( Δ k n , X ) C n Sus  ( X ) := Cor k ⁡ Δ k n , X C_(n)^("Sus ")(X):=Cor_(k)(Delta_(k)^(n),X)C_{n}^{\text {Sus }}(X):=\operatorname{Cor}_{k}\left(\Delta_{k}^{n}, X\right)CnSus (X):=Cork⁡(Δkn,X); the structure of Δ k Δ k ∗ Delta_(k)^(**)\Delta_{k}^{*}Δk∗ as smooth cosimplicial scheme makes n C n Sus ( X ) n ↦ C n Sus  ( X ) n|->C_(n)^("Sus ")(X)n \mapsto C_{n}^{\text {Sus }}(X)n↦CnSus (X) a simplicial abelian group. As above, we have the associated complex C Sus ( X ) C ∗ Sus  ( X ) C_(**)^("Sus ")(X)C_{*}^{\text {Sus }}(X)C∗Sus (X), the Suslin complex of X X XXX, whose homology is the Suslin homology of X X XXX :
H n S u s ( X , Z ) := π n ( | m C m S u s ( X ) | ) = H n ( C S u s ( X ) ) H n S u s ( X , Z ) := Ï€ n m ↦ C m S u s ( X ) = H n C ∗ S u s ( X ) H_(n)^(Sus)(X,Z):=pi_(n)(|m|->C_(m)^(Sus)(X)|)=H_(n)(C_(**)^(Sus)(X))H_{n}^{\mathrm{Sus}}(X, \mathbb{Z}):=\pi_{n}\left(\left|m \mapsto C_{m}^{\mathrm{Sus}}(X)\right|\right)=H_{n}\left(C_{*}^{\mathrm{Sus}}(X)\right)HnSus(X,Z):=Ï€n(|m↦CmSus(X)|)=Hn(C∗Sus(X))
In fact, the monoid of the N N N\mathbb{N}N-linear combinations of irreducible correspondences W X × Y W ⊂ X × Y W sub X xx YW \subset X \times YW⊂X×Y is the same as the monoid of morphisms
ϕ : X n 0 Sym n Y Ï• : X → ⨆ n ≥ 0   Sym n ⁡ Y phi:X rarr⨆_(n >= 0)Sym^(n)Y\phi: X \rightarrow \bigsqcup_{n \geq 0} \operatorname{Sym}^{n} YÏ•:X→⨆n≥0Symn⁡Y
where Sym n Y Sym n ⁡ Y Sym^(n)Y\operatorname{Sym}^{n} YSymn⁡Y is the quotient Y n / Σ n Y n / Σ n Y^(n)//Sigma_(n)Y^{n} / \Sigma_{n}Yn/Σn of Y n Y n Y^(n)Y^{n}Yn by the symmetric group permuting the factors, with the monoid structure induced by the sum map
Sym n Y × Sym m Y Sym m + n Y Sym n ⁡ Y × Sym m ⁡ Y → Sym m + n ⁡ Y Sym^(n)Y xxSym^(m)Y rarrSym^(m+n)Y\operatorname{Sym}^{n} Y \times \operatorname{Sym}^{m} Y \rightarrow \operatorname{Sym}^{m+n} YSymn⁡Y×Symm⁡Y→Symm+n⁡Y
Suslin's complex and his definition of algebraic homology can thus be thought of as an algebraic incarnation of the theorem of Dold-Thom [34, sATz 6.4], that identifies the homotopy groups of the infinite symmetric product of a pointed C W C W CW\mathrm{CW}CW complex T T TTT with the
reduced homology of T T TTT. The main result of [112] gives an isomorphism of the mod n n nnn Suslin homology, H Sus ( X , Z / n ) H ∗ Sus  ( X , Z / n ) H_(**)^("Sus ")(X,Z//n)H_{*}^{\text {Sus }}(X, \mathbb{Z} / n)H∗Sus (X,Z/n), for X X XXX of finite type over C C C\mathbb{C}C, with the mod n mod n mod n\bmod nmodn singular homology of X ( C ) X ( C ) X(C)X(\mathbb{C})X(C), a first major success of the theory.
Let Δ top n Δ top  n Delta_("top ")^(n)\Delta_{\text {top }}^{n}Δtop n denote the usual n n nnn-simplex
Δ top n := { ( t 0 , , t n ) R n + 1 i t i = 1 , t i 0 } Δ top  n := t 0 , … , t n ∈ R n + 1 ∣ ∑ i   t i = 1 , t i ≥ 0 Delta_("top ")^(n):={(t_(0),dots,t_(n))inR^(n+1)∣sum_(i)t_(i)=1,t_(i) >= 0}\Delta_{\text {top }}^{n}:=\left\{\left(t_{0}, \ldots, t_{n}\right) \in \mathbb{R}^{n+1} \mid \sum_{i} t_{i}=1, t_{i} \geq 0\right\}Δtop n:={(t0,…,tn)∈Rn+1∣∑iti=1,ti≥0}
with the inclusion Δ top n Δ n ( C ) Δ top  n ⊂ Δ n ( C ) Delta_("top ")^(n)subDelta^(n)(C)\Delta_{\text {top }}^{n} \subset \Delta^{n}(\mathbb{C})Δtop n⊂Δn(C).
Theorem 2.1 ([112, THEOREM 8.3]). Let X X XXX be separated finite type scheme over C C C\mathbb{C}C and let n 2 n ≥ 2 n >= 2n \geq 2n≥2 be an integer. Then the map
Hom ( Δ C , d 0 Sym d X ) Hom top ( Δ top , d 0 Sym d X ( C ) ) Hom ⁡ Δ C ∗ , ⨆ d ≥ 0   Sym d ⁡ X → Hom top  ⁡ Δ top  ∗ , ⨆ d ≥ 0   Sym d ⁡ X ( C ) Hom(Delta_(C)^(**),⨆_(d >= 0)Sym^(d)X)rarrHom_("top ")(Delta_("top ")^(**),⨆_(d >= 0)Sym^(d)X(C))\operatorname{Hom}\left(\Delta_{\mathbb{C}}^{*}, \bigsqcup_{d \geq 0} \operatorname{Sym}^{d} X\right) \rightarrow \operatorname{Hom}_{\text {top }}\left(\Delta_{\text {top }}^{*}, \bigsqcup_{d \geq 0} \operatorname{Sym}^{d} X(\mathbb{C})\right)Hom⁡(ΔC∗,⨆d≥0Symd⁡X)→Homtop ⁡(Δtop ∗,⨆d≥0Symd⁡X(C))
induced by the inclusions Δ top m Δ m ( C ) Δ top  m ⊂ Δ m ( C ) Delta_("top ")^(m)subDelta^(m)(C)\Delta_{\text {top }}^{m} \subset \Delta^{m}(\mathbb{C})Δtop m⊂Δm(C) gives rise to an isomorphism H Sus ( X , Z / n ) H ∗ Sus  ( X , Z / n ) → H_(**)^("Sus ")(X,Z//n)rarrH_{*}^{\text {Sus }}(X, \mathbb{Z} / n) \rightarrowH∗Sus (X,Z/n)→ H sing ( X ( C ) , Z / n ) H ∗ sing  ( X ( C ) , Z / n ) H_(**)^("sing ")(X(C),Z//n)H_{*}^{\text {sing }}(X(\mathbb{C}), \mathbb{Z} / n)H∗sing (X(C),Z/n).
There is also a corresponding statement for X X XXX over an arbitrary algebraically closed field k k kkk of characteristic zero in terms of étale cohomology [112, THEOREM 7.8]; this extends to characteristic p > 0 p > 0 p > 0p>0p>0 and n n nnn prime to p p ppp by using alterations.

2.3. Quillen-Lichtenbaum conjectures

Quillen's computation of the higher algebraic K K KKK-theory of finite fields and of number rings led to a search for a relation of higher algebraic K K KKK-theory with special values of zeta-functions and L L LLL-functions. We will not go into this in detail here, but to large part, this was responsible for the Beilinson-Lichtenbaum conjectures on the existence of motivic complexes computing the conjectural motivic cohomology. Going back to K K KKK-theory, this suggested that algebraic K K KKK-theory with mod- ℓ ℓ\ellℓ coefficients should be the same as mod- ℓ ℓ\ellℓ étale K K KKK-theory (a purely algebraic version of mod- ℓ ℓ\ellℓ topological K K KKK-theory, see [35]), at least in large enough degrees. This is more precisely stated as the Quillen-Lichtenbaum conjecture
Conjecture 2.2 ([101], [42, CONJEcture 3.9]). Let ℓ ℓ\ellℓ be a prime and let X X XXX be a regular, noetherian scheme with ℓ ℓ\ellℓ invertible on X X XXX. Suppose X X XXX has finite ℓ ℓ\ellℓ-étale cohomological dimension cd ( X ) cd ℓ ⁡ ( X ) cd_(ℓ)(X)\operatorname{cd}_{\ell}(X)cdℓ⁡(X). Then the canonical map
K n ( X ; Z / r ) K n e ́ t ( X ; Z / r ) K n X ; Z / â„“ r → K n e ́ t X ; Z / â„“ r K_(n)(X;Z//â„“^(r))rarrK_(n)^(ét)(X;Z//â„“^(r))K_{n}\left(X ; \mathbb{Z} / \ell^{r}\right) \rightarrow K_{n}^{e ́ t}\left(X ; \mathbb{Z} / \ell^{r}\right)Kn(X;Z/â„“r)→Knét(X;Z/â„“r)
is an isomorphism for n cd ( X ) 1 n ≥ cd ℓ ⁡ ( X ) − 1 n >= cd_(ℓ)(X)-1n \geq \operatorname{cd}_{\ell}(X)-1n≥cdℓ⁡(X)−1 and is injective for n = cd ( X ) 2 n = cd ℓ ⁡ ( X ) − 2 n=cd_(ℓ)(X)-2n=\operatorname{cd}_{\ell}(X)-2n=cdℓ⁡(X)−2.
Here K n et ( X ; Z / ) K n et  ( X ; Z / ℓ ) K_(n)^("et ")(X;Z//ℓ)K_{n}^{\text {et }}(X ; \mathbb{Z} / \ell)Knet (X;Z/ℓ) is the étale K K KKK-theory developed by Dwyer and Friedlander [35, 41,42].
Conjecture 2.2 for a smooth k k kkk-scheme is essentially a consequence of the BeilinsonLichtenbaum axioms (without Beilinson-Soulé vanishing). The Beilinson-Lichtenbaum conjecture (iv)(a,b) says that the comparison map Γ ( q ) / r R α μ r q Γ ( q ) / â„“ r → R α ∗ μ â„“ r ⊗ q Gamma(q)//â„“^(r)rarr Ralpha_(**)mu_(â„“^(r))^(ox q)\Gamma(q) / \ell^{r} \rightarrow R \alpha_{*} \mu_{\ell^{r}}^{\otimes q}Γ(q)/â„“r→Rα∗μℓr⊗q induces an isomorphism on cohomology sheaves up to degree q q qqq. Combining the local-global spectral sequence
for some X S m k X ∈ S m k X inSm_(k)X \in \mathrm{Sm}_{k}X∈Smk with the Atiyah-Hirzebruch spectral sequences from motivic cohomology to K K KKK-theory (axiom (v)) and from étale cohomology to étale K K KKK-theory, and keeping track of the cohomological bound in the Beilinson-Lichtenbaum conjecture gives the result.

2.4. Voevodsky's category DM and modern motivic cohomology

One can almost realize Beilinson's ideas of a categorical framework for motivic cohomology by working in the setting of triangulated categories, viewed as a replacement for the derived category of Beilinson's conjectured abelian category of motivic sheaves. Once this is accomplished, one could hope that an abelian category of mixed motives could be constructed out of the triangulated category as the heart of a suitable t t ttt-structure.
Constructions of a triangulated category of mixed motives over a perfect base-field were given by Hanamura [57-59], Voevodsky [127], and myself [83]. All three categories yield Bloch's higher Chow groups as the categorical motivic cohomology, however, Voevodsky's sheaf-theoretic approach has had the most far-reaching consequences and has been widely adopted as the correct solution. The construction of a motivic t t ttt-structure is still an open problem. 1 1 ^(1){ }^{1}1 There are also constructions of triangulated categories of mixed motives by the method of compatible realizations, such as by Huber [64], or Nori's construction of an abelian category of mixed motives, described in [65, PART II]; we will not pursue these directions here. We also refer the reader to Jannsen's survey on mixed motives [68].
Voevodsky's triangulated category of motives over k , DM ( k ) k , DM ⁡ ( k ) k,DM(k)k, \operatorname{DM}(k)k,DM⁡(k), is based on the category of finite correspondences on S m k S m k Sm_(k)\mathrm{Sm}_{k}Smk, a refinement of Grothendieck's composition law for correspondences on smooth projective varieties. Grothendieck had constructed categories of motives for smooth projective varieties, with the morphisms from X X XXX to Y Y YYY given by the group of cycles modulo rational equivalence C H dim X ( X × Y ) C H dim ⁡ X ( X × Y ) CH_(dim X)(X xx Y)\mathrm{CH}_{\operatorname{dim} X}(X \times Y)CHdim⁡X(X×Y). The composition law is given by
(2.1) W W = p X Z ( p X Y ( W ) p Y Z ( W ) ) (2.1) W ′ ∘ W = p X Z ∗ p X Y ∗ ( W ) ⋅ p Y Z ∗ W ′ {:(2.1)W^(')@W=p_(XZ**)(p_(XY)^(**)(W)*p_(YZ)^(**)(W^('))):}\begin{equation*} W^{\prime} \circ W=p_{X Z *}\left(p_{X Y}^{*}(W) \cdot p_{Y Z}^{*}\left(W^{\prime}\right)\right) \tag{2.1} \end{equation*}(2.1)W′∘W=pXZ∗(pXY∗(W)⋅pYZ∗(W′))
with W C H dim X ( X × Y ) W ∈ C H dim ⁡ X ( X × Y ) W inCH_(dim X)(X xx Y)W \in \mathrm{CH}_{\operatorname{dim} X}(X \times Y)W∈CHdim⁡X(X×Y) and W C H dim Y ( Y × Z ) W ′ ∈ C H dim ⁡ Y ( Y × Z ) W^(')inCH_(dim Y)(Y xx Z)W^{\prime} \in \mathrm{CH}_{\operatorname{dim} Y}(Y \times Z)W′∈CHdim⁡Y(Y×Z); one needs to pass to cycle classes to define p X Y ( W ) p Y Z ( W ) p X Y ∗ ( W ) â‹… p Y Z ∗ W ′ p_(XY)^(**)(W)*p_(YZ)^(**)(W^('))p_{X Y}^{*}(W) \cdot p_{Y Z}^{*}\left(W^{\prime}\right)pXY∗(W)â‹…pYZ∗(W′) and the projection p X Z p X Z p_(XZ)p_{X Z}pXZ needs to be proper (that is, Y Y YYY needs to be proper over k k kkk ) to define p X Z p X Z ∗ p_(XZ**)p_{X Z *}pXZ∗.
Voevodsky's key insight was to restrict to finite correspondences, so that all the operations used in the composition law of correspondence classes would be defined on the level of the cycles themselves, without needing to pass to rational equivalence classes, and without needing the varieties involved to be proper. Voevodsky's idea of having a well-defined composition law on a restricted class of correspondences has been modified and applied in a wide range of different contexts, somewhat similar to the use of various flavors of bordism theories in topology.
Let X X XXX and Y Y YYY be in Sm k Sm k Sm_(k)\operatorname{Sm}_{k}Smk. Recall from Section 2.2 the subgroup Cor k ( X , Y ) Cor k ⁡ ( X , Y ) ⊂ Cor_(k)(X,Y)sub\operatorname{Cor}_{k}(X, Y) \subsetCork⁡(X,Y)⊂ Z dim X ( X × Y ) Z dim ⁡ X ( X × Y ) Z_(dim X)(X xx Y)Z_{\operatorname{dim} X}(X \times Y)Zdim⁡X(X×Y) generated by the integral W X × Y W ⊂ X × Y W sub X xx YW \subset X \times YW⊂X×Y that are finite over X X XXX and map surjectively to a component of X X XXX.
Lemma 2.3. Let X , Y , Z X , Y , Z X,Y,ZX, Y, ZX,Y,Z be smooth k k kkk-varieties and take α Cor k ( X , Y ) , β Cor k ( Y , Z ) α ∈ Cor k ⁡ ( X , Y ) , β ∈ Cor k ⁡ ( Y , Z ) alpha inCor_(k)(X,Y),beta inCor_(k)(Y,Z)\alpha \in \operatorname{Cor}_{k}(X, Y), \beta \in \operatorname{Cor}_{k}(Y, Z)α∈Cork⁡(X,Y),β∈Cork⁡(Y,Z). Then
(i) The cycles p Y Z ( β ) p Y Z ∗ ( β ) p_(YZ)^(**)(beta)p_{Y Z}^{*}(\beta)pYZ∗(β) and p X Y ( α ) p X Y ∗ ( α ) p_(XY)^(**)(alpha)p_{X Y}^{*}(\alpha)pXY∗(α) intersect properly on X × Y × Z X × Y × Z X xx Y xx ZX \times Y \times ZX×Y×Z, so the inter section product p Y Z ( β ) p X Y ( α ) p Y Z ∗ ( β ) â‹… p X Y ∗ ( α ) p_(YZ)^(**)(beta)*p_(XY)^(**)(alpha)p_{Y Z}^{*}(\beta) \cdot p_{X Y}^{*}(\alpha)pYZ∗(β)â‹…pXY∗(α) exists as a well-defined cycle on X × Y × Z X × Y × Z X xx Y xx ZX \times Y \times ZX×Y×Z.
(ii) Letting | α | X × Y | α | ⊂ X × Y |alpha|sub X xx Y|\alpha| \subset X \times Y|α|⊂X×Y, and | β | Y × Z | β | ⊂ Y × Z |beta|sub Y xx Z|\beta| \subset Y \times Z|β|⊂Y×Z denote the support of α α alpha\alphaα and β β beta\betaβ, respectively, each irreducible component of the intersection X × | β | | α | × Z X × | β | ∩ | α | × Z X xx|beta|nn|alpha|xx ZX \times|\beta| \cap|\alpha| \times ZX×|β|∩|α|×Z is finite over X × Z X × Z X xx ZX \times ZX×Z, and maps surjectively onto some component of X X XXX.
In other words, the formula
β α = p X Z ( p Y Z β p X Y α ) β ∘ α = p X Z ∗ p Y Z ∗ β â‹… p X Y ∗ α beta@alpha=p_(XZ**)(p_(YZ)^(**)beta*p_(XY)^(**)alpha)\beta \circ \alpha=p_{X Z *}\left(p_{Y Z}^{*} \beta \cdot p_{X Y}^{*} \alpha\right)β∘α=pXZ∗(pYZ∗β⋅pXY∗α)
makes sense for α Cor k ( X , Y ) α ∈ Cor k ⁡ ( X , Y ) alpha inCor_(k)(X,Y)\alpha \in \operatorname{Cor}_{k}(X, Y)α∈Cork⁡(X,Y) and β Cor k ( Y , Z ) β ∈ Cor k ⁡ ( Y , Z ) beta inCor_(k)(Y,Z)\beta \in \operatorname{Cor}_{k}(Y, Z)β∈Cork⁡(Y,Z), and the resulting cycle on X × Z X × Z X xx ZX \times ZX×Z is in Cor k ( X , Z ) Cor k ⁡ ( X , Z ) Cor_(k)(X,Z)\operatorname{Cor}_{k}(X, Z)Cork⁡(X,Z). This defines the composition law in Voevodsky's category of finite correspondences, Cor k Cor k Cor_(k)\operatorname{Cor}_{k}Cork, with objects as for Sm k Sm k Sm_(k)\operatorname{Sm}_{k}Smk, and morphisms Hom Cor k ( X , Y ) = Cor k ( X , Y ) Hom Cor k ⁡ ( X , Y ) = Cor k ⁡ ( X , Y ) Hom_(Cor_(k))(X,Y)=Cor_(k)(X,Y)\operatorname{Hom}_{\operatorname{Cor}_{k}}(X, Y)=\operatorname{Cor}_{k}(X, Y)HomCork⁡(X,Y)=Cork⁡(X,Y). Sending a usual morphism f : X Y f : X → Y f:X rarr Yf: X \rightarrow Yf:X→Y of smooth varieties to its graph defines a faithful functor [ ] : S m k [ − ] : S m k → [-]:Sm_(k)rarr[-]: \mathrm{Sm}_{k} \rightarrow[−]:Smk→ Cor k k _(k)_{k}k.
Once one has the category C o r k C o r k Cor_(k)\mathrm{Cor}_{k}Cork, the path to D M ( k ) D M ( k ) DM(k)\mathrm{DM}(k)DM(k) is easy to describe. One takes the category of additive presheaves of abelian groups on C o r k C o r k Cor_(k)\mathrm{Cor}_{k}Cork, the category of presheaves with transfer PST ( k ) PST ⁡ ( k ) PST(k)\operatorname{PST}(k)PST⁡(k). Inside PST ( k ) PST ⁡ ( k ) PST(k)\operatorname{PST}(k)PST⁡(k) is the category NST ( k ) ( k ) (k)(k)(k) of Nisnevich sheaves with transfer, that is, a presheaf that is a Nisnevich sheaf when restricted to Sm k Cor k Sm k ⊂ Cor k Sm_(k)subCor_(k)\operatorname{Sm}_{k} \subset \operatorname{Cor}_{k}Smk⊂Cork. Each X Sm k X ∈ Sm k X inSm_(k)X \in \operatorname{Sm}_{k}X∈Smk defines an object Z t r ( X ) NST ( k ) Z t r ( X ) ∈ NST ⁡ ( k ) Z_(tr)(X)in NST(k)\mathbb{Z}_{\mathrm{tr}}(X) \in \operatorname{NST}(k)Ztr(X)∈NST⁡(k), as the representable (pre)sheaf Y Cor k ( Y , X ) Y ↦ Cor k ⁡ ( Y , X ) Y|->Cor_(k)(Y,X)Y \mapsto \operatorname{Cor}_{k}(Y, X)Y↦Cork⁡(Y,X). Inside the derived category D ( NST ( k ) ) D ( NST ⁡ ( k ) ) D(NST(k))D(\operatorname{NST}(k))D(NST⁡(k)) is the full subcategory of complexes K K KKK whose homology presheaves h _ i ( K ) h _ i ( K ) h__(i)(K)\underline{h}_{i}(K)h_i(K) are A 1 A 1 A^(1)\mathbb{A}^{1}A1-homotopy invariant: h _ i ( K ) ( X ) h _ i ( K ) ( X × A 1 ) h _ i ( K ) ( X ) ≅ h _ i ( K ) X × A 1 h__(i)(K)(X)~=h__(i)(K)(X xxA^(1))\underline{h}_{i}(K)(X) \cong \underline{h}_{i}(K)\left(X \times \mathbb{A}^{1}\right)h_i(K)(X)≅h_i(K)(X×A1) for all X S m k X ∈ S m k X inSm_(k)X \in \mathrm{Sm}_{k}X∈Smk. This is the category of effective motives D M eff ( k ) D M eff  ( k ) DM^("eff ")(k)\mathrm{DM}^{\text {eff }}(k)DMeff (k). The Suslin complex construction, P C Sus ( P ) P ↦ C ∗ Sus  ( P ) P|->C_(**)^("Sus ")(P)\mathcal{P} \mapsto C_{*}^{\text {Sus }}(\mathcal{P})P↦C∗Sus (P), with
C S u s ( P ) ( X ) := P ( X × Δ ) C ∗ S u s ( P ) ( X ) := P X × Δ ∗ C_(**)^(Sus)(P)(X):=P(X xxDelta^(**))C_{*}^{\mathrm{Sus}}(\mathcal{P})(X):=\mathcal{P}\left(X \times \Delta^{*}\right)C∗Sus(P)(X):=P(X×Δ∗)
extends to a functor R C Sus : D ( N S T ( k ) ) D M eff ( k ) R C ∗ Sus  : D ( N S T ( k ) ) → D M eff  ( k ) RC_(**)^("Sus "):D(NST(k))rarrDM^("eff ")(k)R C_{*}^{\text {Sus }}: D(\mathrm{NST}(k)) \rightarrow \mathrm{DM}^{\text {eff }}(k)RC∗Sus :D(NST(k))→DMeff (k), and realizes D M eff ( k ) D M eff  ( k ) DM^("eff ")(k)\mathrm{DM}^{\text {eff }}(k)DMeff (k) as the localization of D ( NST ( k ) ) D ( NST ⁡ ( k ) ) D(NST(k))D(\operatorname{NST}(k))D(NST⁡(k)) with respect to the complexes Z t r ( X × A 1 ) p Z t r ( X ) Z t r X × A 1 → p ∗ Z t r ( X ) Z_(tr)(X xxA^(1))rarr"p_(**)"Z_(tr)(X)\mathbb{Z}_{\mathrm{tr}}\left(X \times \mathbb{A}^{1}\right) \xrightarrow{p_{*}} \mathbb{Z}_{\mathrm{tr}}(X)Ztr(X×A1)→p∗Ztr(X). Via R C Sus R C ∗ Sus  RC_(**)^("Sus ")R C_{*}^{\text {Sus }}RC∗Sus  D M e f f ( k ) D M e f f ( k ) DM^(eff)(k)\mathrm{DM}^{\mathrm{eff}}(k)DMeff(k) inherits a tensor structure ⊗ ox\otimes⊗ from D ( N S T ( k ) ) D ( N S T ( k ) ) D(NST(k))D(\mathrm{NST}(k))D(NST(k)). The functor Z t r : S m k N S T ( k ) Z t r : S m k → N S T ( k ) Z_(tr):Sm_(k)rarrNST(k)\mathbb{Z}_{\mathrm{tr}}: \mathrm{Sm}_{k} \rightarrow \mathrm{NST}(k)Ztr:Smk→NST(k) defines the functor M eff := R C S u s Z t r M eff  := R C ∗ S u s ∘ Z t r M^("eff "):=RC_(**)^(Sus)@Z_(tr)M^{\text {eff }}:=R C_{*}^{\mathrm{Sus}} \circ \mathbb{Z}_{\mathrm{tr}}Meff :=RC∗Sus∘Ztr,
M e f f : S m k D M e f f ( k ) M e f f : S m k → D M e f f ( k ) M^(eff):Sm_(k)rarrDM^(eff)(k)M^{\mathrm{eff}}: \mathrm{Sm}_{k} \rightarrow \mathrm{DM}^{\mathrm{eff}}(k)Meff:Smk→DMeff(k)
The Tate object Z ( 1 ) D M e f f ( k ) Z ( 1 ) ∈ D M e f f ( k ) Z(1)inDM^(eff)(k)\mathbb{Z}(1) \in \mathrm{DM}^{\mathrm{eff}}(k)Z(1)∈DMeff(k) is the image of the complex Z t r ( Z t r ( Z_(tr)(\mathbb{Z}_{\mathrm{tr}}(Ztr( Spec k ) i k ) → i ∞ ∗ k)rarr"i_(oo**)"k) \xrightarrow{i_{\infty *}}k)→i∞∗ Z t r ( P 1 ) Z t r P 1 Z_(tr)(P^(1))\mathbb{Z}_{\mathrm{tr}}\left(\mathbb{P}^{1}\right)Ztr(P1) (with Z t r ( P 1 ) Z t r P 1 Z_(tr)(P^(1))\mathbb{Z}_{\mathrm{tr}}\left(\mathbb{P}^{1}\right)Ztr(P1) in degree 2 ) via R C Sus R C ∗ Sus  RC_(**)^("Sus ")R C_{*}^{\text {Sus }}RC∗Sus . One forms the triangulated tensor category DM ( k ) DM ⁡ ( k ) DM(k)\operatorname{DM}(k)DM⁡(k) as the category of Z ( 1 ) − ⊗ Z ( 1 ) -oxZ(1)-\otimes \mathbb{Z}(1)−⊗Z(1)-spectrum objects in D M eff ( k ) D M eff  ( k ) DM^("eff ")(k)\mathrm{DM}^{\text {eff }}(k)DMeff (k), inverting the endofunctor Z ( 1 ) − ⊗ Z ( 1 ) -oxZ(1)-\otimes \mathbb{Z}(1)−⊗Z(1); for M D M ( k ) M ∈ D M ( k ) M inDM(k)M \in \mathrm{DM}(k)M∈DM(k), one has the Tate twists M ( n ) := M Z ( 1 ) n M ( n ) := M ⊗ Z ( 1 ) ⊗ n M(n):=M ox Z(1)^(ox n)M(n):=M \otimes Z(1)^{\otimes n}M(n):=M⊗Z(1)⊗n for n Z n ∈ Z n inZn \in \mathbb{Z}n∈Z; in particular, we have the Tate objects Z ( n ) Z ( n ) Z(n)\mathbb{Z}(n)Z(n). The functor M eff M eff  M^("eff ")M^{\text {eff }}Meff  induces the functor M : S m k D M ( k ) M : S m k → D M ( k ) M:Sm_(k)rarrDM(k)M: \mathrm{Sm}_{k} \rightarrow \mathrm{DM}(k)M:Smk→DM(k).
Bloch's higher Chow groups, Suslin homology, and the motivic complexes Z B I ( q ) Z B I ( q ) Z_(BI)(q)\mathbb{Z}_{\mathrm{BI}}(q)ZBI(q) are represented in D M ( k ) D M ( k ) DM(k)\mathrm{DM}(k)DM(k) via canonical isomorphisms
C H q ( X , 2 q p ) = H p ( X Z a r , Z B l ( q ) ) Hom D M ( k ) ( M ( X ) , Z ( q ) [ p ] ) H n S u s ( X , Z ) = H n ( C S u s ( X ) ) Hom D M ( k ) ( Z [ n ] , M ( X ) ) C H q ( X , 2 q − p ) = H p X Z a r , Z B l ( q ) ≅ Hom D M ( k ) ⁡ ( M ( X ) , Z ( q ) [ p ] ) H n S u s ( X , Z ) = H n C ∗ S u s ( X ) ≅ Hom D M ( k ) ⁡ ( Z [ n ] , M ( X ) ) {:[CH^(q)(X","2q-p)=H^(p)(X_(Zar),Z_(Bl)(q))~=Hom_(DM(k))(M(X)","Z(q)[p])],[H_(n)^(Sus)(X","Z)=H_(n)(C_(**)^(Sus)(X))~=Hom_(DM(k))(Z[n]","M(X))]:}\begin{aligned} \mathrm{CH}^{q}(X, 2 q-p) & =\mathbb{H}^{p}\left(X_{\mathrm{Zar}}, \mathbb{Z}_{\mathrm{Bl}}(q)\right) \cong \operatorname{Hom}_{\mathrm{DM}(k)}(M(X), \mathbb{Z}(q)[p]) \\ H_{n}^{\mathrm{Sus}}(X, \mathbb{Z}) & =H_{n}\left(C_{*}^{\mathrm{Sus}}(X)\right) \cong \operatorname{Hom}_{\mathrm{DM}(k)}(\mathbb{Z}[n], M(X)) \end{aligned}CHq(X,2q−p)=Hp(XZar,ZBl(q))≅HomDM(k)⁡(M(X),Z(q)[p])HnSus(X,Z)=Hn(C∗Sus(X))≅HomDM(k)⁡(Z[n],M(X))
In addition, one has the presheaf of complexes Z V ( q ) Z V ( q ) Z_(V)(q)\mathbb{Z}_{V}(q)ZV(q) on S m k S m k Sm_(k)\mathrm{Sm}_{k}Smk
Z V ( q ) ( X ) := C S u s ( Z t r ( G m ) q [ q ] ) ( X ) Z V ( q ) ( X ) := C − ∗ S u s Z t r G m ⊗ q [ − q ] ( X ) Z_(V)(q)(X):=C_(-**)^(Sus)(Z_(tr)(G_(m))^(ox q)[-q])(X)\mathbb{Z}_{V}(q)(X):=C_{-*}^{\mathrm{Sus}}\left(\mathbb{Z}_{\mathrm{tr}}\left(\mathbb{G}_{m}\right)^{\otimes q}[-q]\right)(X)ZV(q)(X):=C−∗Sus(Ztr(Gm)⊗q[−q])(X)
where Z t r ( G m ) Z t r G m Z_(tr)(G_(m))\mathbb{Z}_{\mathrm{tr}}\left(\mathbb{G}_{m}\right)Ztr(Gm) is the quotient presheaf Z t r ( A 1 { 0 } ) / Z t r ( { 1 } ) Z t r A 1 ∖ { 0 } / Z t r ( { 1 } ) Z_(tr)(A^(1)\\{0})//Z_(tr)({1})\mathbb{Z}_{\mathrm{tr}}\left(\mathbb{A}^{1} \backslash\{0\}\right) / \mathbb{Z}_{\mathrm{tr}}(\{1\})Ztr(A1∖{0})/Ztr({1}). The complexes Z V ( q ) Z V ( q ) Z_(V)(q)\mathbb{Z}_{V}(q)ZV(q) and Z B I ( q ) Z B I ( q ) Z_(BI)(q)\mathbb{Z}_{\mathrm{BI}}(q)ZBI(q) define isomorphic objects in D M e f f ( k ) D M e f f ( k ) DM^(eff)(k)\mathrm{DM}^{\mathrm{eff}}(k)DMeff(k), in particular, are isomorphic in the derived category of Nisnevich sheaves on S m k S m k Sm_(k)\mathrm{Sm}_{k}Smk. The details of these constructions and results are carried out in [127] (with a bit of help from [117]).

2.5. Motivic homotopy theory

Although Voevodsky's triangulated category of motives does give motivic cohomology a categorical foundation, this is really a halfway station on the way to a really suitable categorical framework. As analogy, embedding the Beilinson-Lichtenbaum/Bloch-Suslin theory of motivic complexes in D M ( k ) D M ( k ) DM(k)\mathrm{DM}(k)DM(k) is like considering the singular chain or cochain complex of a topological space as an object in the derived category of abelian groups. A much more fruitful framework for singular (co)homology is to be found in the stable homotopy category SH.
A parallel representability for motivic cohomology for schemes over a base-scheme B B BBB in a wider category of good cohomology theories is to be found in the motivic stable homotopy category over B , S H ( B ) B , S H ( B ) B,SH(B)B, \mathrm{SH}(B)B,SH(B). This, together with the motivic unstable homotopy category, H ( B ) H ( B ) H(B)\mathscr{H}(B)H(B), gives the proper setting for the deeper study of motivic cohomology, besides placing this theory on a equal footing with all cohomology theories on algebraic varieties that satisfy a few natural axioms.
Just as the category D M ( k ) D M ( k ) DM(k)\mathrm{DM}(k)DM(k) starts out as a category of presheaves, the category S H ( B ) S H ( B ) SH(B)\mathrm{SH}(B)SH(B) starts out with the category of presheaves of simplicial sets on S m B S m B Sm_(B)\mathrm{Sm}_{B}SmB. The construction of the unstable motivic homotopy category H ( B ) H ( B ) H(B)\mathscr{H}(B)H(B) over a general base-scheme B B BBB as a suitable localization of this presheaf category was achieved by Morel-Voevodsky [94] and the stable version S H ( B ) S H ( B ) SH(B)\mathrm{SH}(B)SH(B) was described by Voevodsky in his ICM address [116]. The important sixfunctor formalism of Grothendieck was sketched out by Voevodsky and realized in detail by Ayoub [ 5 , 6 ] [ 5 , 6 ] [5,6][5,6][5,6]. A general theory of motivic categories with a six-functor formalism, including S H ( ) S H ( − ) SH(-)\mathrm{SH}(-)SH(−), was established by Cisinski-Déglise [33], and Hoyois [62] gave a construction on the level of infinity categories for an equivariant version. A new point of view, the approach of framed correspondences, also first sketched by Voevodsky [126], is a breakthrough in our understanding of the infinite loop objects in the motivic setting, and concerning our main interest, motivic cohomology, has led to a natural construction of motivic cohomology over a general base-scheme.
In topology, the representation of singular (co)homology via the singular (co)chain complexes is placed in the setting of stable homotopy theory through the construction of the
Eilenberg-MacLane spectra, giving a natural isomorphism for each abelian group A A AAA,
H n ( X , A ) Hom S H ( Σ X + , Σ n EM ( A ) ) H n ( X , A ) ≅ Hom S H ⁡ Σ ∞ X + , Σ n EM ⁡ ( A ) H^(n)(X,A)~=Hom_(SH)(Sigma^(oo)X_(+),Sigma^(n)EM(A))H^{n}(X, A) \cong \operatorname{Hom}_{\mathrm{SH}}\left(\Sigma^{\infty} X_{+}, \Sigma^{n} \operatorname{EM}(A)\right)Hn(X,A)≅HomSH⁡(Σ∞X+,ΣnEM⁡(A))
with the Eilenberg-MacLane spectrum EM ( A ) EM ⁡ ( A ) EM(A)\operatorname{EM}(A)EM⁡(A) being characterized by its stable homotopy groups
π n s ( EM ( A ) ) = { A for n = 0 0 else Ï€ n s ( EM ⁡ ( A ) ) = A  for  n = 0 0  else  pi_(n)^(s)(EM(A))={[A," for "n=0],[0," else "]:}\pi_{n}^{s}(\operatorname{EM}(A))= \begin{cases}A & \text { for } n=0 \\ 0 & \text { else }\end{cases}Ï€ns(EM⁡(A))={A for n=00 else 
The assignment A EM ( A ) A ↦ EM ⁡ ( A ) A|->EM(A)A \mapsto \operatorname{EM}(A)A↦EM⁡(A) extends to a fully faithful embedding EM : D ( A b ) S H D ( A b ) → S H D(Ab)rarrSHD(\mathbf{A b}) \rightarrow \mathrm{SH}D(Ab)→SH. This realizes the ordinary (co)homology as being represented by the derived category D ( A b ) D ( A b ) D(Ab)D(\mathbf{A b})D(Ab) via its Eilenberg-MacLane embedding in SH, which in turn is to be viewed as the category of all cohomology theories on reasonable topological spaces.
The stable homotopy category SH is the stabilization of the unstable pointed homotopy category H H ∙ H_(∙)\mathscr{H}_{\bullet}H∙ with respect to the suspension operator Σ X := S 1 X Σ X := S 1 ∧ X Sigma X:=S^(1)^^X\Sigma X:=S^{1} \wedge XΣX:=S1∧X, which becomes an invertible endofunctor on S H S H SH\mathrm{SH}SH. The resulting functor of H H ∙ H_(∙)\mathscr{H}_{\bullet}H∙ to its stabilization is the infinite suspension functor Σ Î£ ∞ Sigma^(oo)\Sigma^{\infty}Σ∞ and gives us the "effective" subcategory S H eff S H S H eff  ⊂ S H SH^("eff ")subSH\mathrm{SH}^{\text {eff }} \subset \mathrm{SH}SHeff ⊂SH, as the smallest subcategory containing Σ ( H ) Σ ∞ H ∙ Sigma^(oo)(H_(∙))\Sigma^{\infty}\left(\mathscr{H}_{\bullet}\right)Σ∞(H∙) and closed under homotopy cofibers and small coproducts. This in turn gives a decreasing filtration on S H S H SH\mathrm{SH}SH by the subcategories Σ n S H e f f Σ n S H e f f Sigma^(n)SH^(eff)\Sigma^{n} \mathrm{SH}^{\mathrm{eff}}ΣnSHeff, n Z n ∈ Z n inZn \in \mathbb{Z}n∈Z. This rather abstract looking filtration is simply the filtration by connectivity: E E EEE is in Σ n S H e f f Σ n S H e f f Sigma^(n)SH^(eff)\Sigma^{n} \mathrm{SH}^{\mathrm{eff}}ΣnSHeff if and only if π m s E = 0 Ï€ m s E = 0 pi_(m)^(s)E=0\pi_{m}^{s} E=0Ï€msE=0 for m < n m < n m < nm<nm<n. The layers in this filtration are isomorphic to the category A b A b Ab\mathbf{A b}Ab, by the functor E π n E E ↦ Ï€ n E E|->pi_(n)EE \mapsto \pi_{n} EE↦πnE, and in fact, this filtration is the one given by a natural t t ttt-structure on S H S H SH\mathrm{SH}SH with heart A b A b Ab\mathbf{A b}Ab; concretely, the 0 th truncation τ 0 E Ï„ 0 E tau_(0)E\tau_{0} EÏ„0E is given by the Eilenberg-MacLane spectrum EM ( π 0 ( E ) ) EM ⁡ Ï€ 0 ( E ) EM(pi_(0)(E))\operatorname{EM}\left(\pi_{0}(E)\right)EM⁡(Ï€0(E)).
A central example is the sphere spectrum S := Σ S 0 S := Σ ∞ S 0 S:=Sigma^(oo)S^(0)\mathbb{S}:=\Sigma^{\infty} S^{0}S:=Σ∞S0. Since
π 0 s S = colim m π m ( S m ) = Z Ï€ 0 s S = colim m ⁡ Ï€ m S m = Z pi_(0)^(s)S=colim_(m)pi_(m)(S^(m))=Z\pi_{0}^{s} \mathbb{S}=\operatorname{colim}_{m} \pi_{m}\left(S^{m}\right)=\mathbb{Z}Ï€0sS=colimm⁡πm(Sm)=Z
we have τ 0 S = EM ( Z ) Ï„ 0 S = EM ⁡ ( Z ) tau_(0)S=EM(Z)\tau_{0} \mathbb{S}=\operatorname{EM}(\mathbb{Z})Ï„0S=EM⁡(Z), establishing the natural relation between homology and homotopy.
In the motivic world, we have a somewhat parallel picture. The pointed unstable category H ( B ) H ∙ ( B ) H_(∙)(B)H_{\bullet}(B)H∙(B) has a natural 2-parameter family of "spheres." Let S n S n S^(n)S^{n}Sn denote the constant presheaf with value the pointed n n nnn-sphere, and let G m G m G_(m)\mathbb{G}_{m}Gm denote the representable presheaf A 1 { 0 } A 1 ∖ { 0 } A^(1)\\{0}\mathbb{A}^{1} \backslash\{0\}A1∖{0} pointed by 1 . Define
S a , b := S a b G m b S a , b := S a − b ∧ G m ∧ b S^(a,b):=S^(a-b)^^G_(m)^(^^b)S^{a, b}:=S^{a-b} \wedge \mathbb{G}_{m}^{\wedge b}Sa,b:=Sa−b∧Gm∧b
for a b 0 a ≥ b ≥ 0 a >= b >= 0a \geq b \geq 0a≥b≥0. We consider P 1 P 1 P^(1)\mathbb{P}^{1}P1 as the representable presheaf, pointed by 1 ; there is a canonical isomorphism P 1 S 2 , 1 P 1 ≅ S 2 , 1 P^(1)~=S^(2,1)\mathbb{P}^{1} \cong S^{2,1}P1≅S2,1 in H ( B ) H ∙ ( B ) H_(∙)(B)\mathscr{H}_{\bullet}(B)H∙(B).
In order to achieve the analog of Spanier-Whitehead duality in the motivic setting, one needs to use spectra with respect to P 1 P 1 P^(1)\mathbb{P}^{1}P1-suspension rather than with respect to S 1 S 1 S^(1)S^{1}S1-suspension. The category S H ( B ) S H ( B ) SH(B)\mathrm{SH}(B)SH(B) is constructed as a homotopy category of P 1 P 1 P^(1)\mathbb{P}^{1}P1-spectra in H ( B ) H ∙ ( B ) H_(∙)(B)\mathscr{H}_{\bullet}(B)H∙(B), so P 1 P 1 P^(1)\mathbb{P}^{1}P1-suspension becomes invertible and our family of spheres extends to a family of invertible suspension endofunctors
Σ a , b : S H ( B ) S H ( B ) , a , b Z Σ a , b : S H ( B ) → S H ( B ) , a , b ∈ Z Sigma^(a,b):SH(B)rarrSH(B),quad a,b inZ\Sigma^{a, b}: \mathrm{SH}(B) \rightarrow \mathrm{SH}(B), \quad a, b \in \mathbb{Z}Σa,b:SH(B)→SH(B),a,b∈Z
Each E S H ( B ) E ∈ S H ( B ) E inSH(B)E \in \mathrm{SH}(B)E∈SH(B) gives the bigraded cohomology theory on S m B S m B Sm_(B)\mathrm{Sm}_{B}SmB by
E a , b ( X ) := Hom S H ( B ) ( Σ P 1 X + , Σ a , b E ) E a , b ( X ) := Hom S H ( B ) ⁡ Σ P 1 ∞ X + , Σ a , b ∧ E E^(a,b)(X):=Hom_(SH(B))(Sigma_(P^(1))^(oo)X_(+),Sigma^(a,b)^^E)E^{a, b}(X):=\operatorname{Hom}_{\mathrm{SH}(B)}\left(\Sigma_{\mathbb{P}^{1}}^{\infty} X_{+}, \Sigma^{a, b} \wedge E\right)Ea,b(X):=HomSH(B)⁡(ΣP1∞X+,Σa,b∧E)
Note that the translation in S H ( B ) S H ( B ) SH(B)\mathrm{SH}(B)SH(B) is given by S 1 S 1 S^(1)S^{1}S1-suspension, not P 1 P 1 P^(1)\mathbb{P}^{1}P1-suspension.
The effective subcategory S H eff ( B ) S H eff  ( B ) SH^("eff ")(B)\mathrm{SH}^{\text {eff }}(B)SHeff (B) is defined as the localizing subcategory (i.e., a triangulated subcategory closed under small coproducts) generated by the P 1 P 1 P^(1)\mathbb{P}^{1}P1-infinite suspension spectra Σ P 1 X Σ P 1 ∞ X Sigma_(P^(1))^(oo)X\Sigma_{\mathbb{P}^{1}}^{\infty} \mathcal{X}ΣP1∞X for X H ( B ) X ∈ H ∙ ( B ) XinH_(∙)(B)\mathcal{X} \in \mathscr{H}_{\bullet}(B)X∈H∙(B). We replace the filtration of S H S H SH\mathrm{SH}SH with respect to S 1 S 1 S^(1)S^{1}S1-connectivity with the filtration on S H ( B ) S H ( B ) SH(B)\mathrm{SH}(B)SH(B) with respect to P 1 P 1 P^(1)\mathbb{P}^{1}P1-connectivity, via the subcategories Σ P 1 1 n S H e f f ( B ) Σ P 1 1 n S H e f f ( B ) Sigma_(P_(1)^(1))^(n)SH^(eff)(B)\Sigma_{\mathbb{P}_{1}^{1}}^{n} \mathrm{SH}^{\mathrm{eff}}(B)ΣP11nSHeff(B). This is Voevodsky's slice filtration, with associated n n nnnth truncation denoted f n f n f_(n)f_{n}fn, giving for each E S H ( B ) E ∈ S H ( B ) E inSH(B)E \in \mathrm{SH}(B)E∈SH(B) the tower
f n + 1 E f n E E ⋯ → f n + 1 E → f n E → ⋯ → E cdots rarrf_(n+1)E rarrf_(n)E rarr cdots rarr E\cdots \rightarrow f_{n+1} E \rightarrow f_{n} E \rightarrow \cdots \rightarrow E⋯→fn+1E→fnE→⋯→E
One has the layers s n E s n E s_(n)Es_{n} EsnE of this tower, fitting into a distinguished triangle
f n + 1 E f n E s n E f n + 1 E [ 1 ] = Σ 1 , 0 f n + 1 E f n + 1 E → f n E → s n E → f n + 1 E [ 1 ] = Σ 1 , 0 f n + 1 E f_(n+1)E rarrf_(n)E rarrs_(n)E rarrf_(n+1)E[1]=Sigma^(1,0)f_(n+1)Ef_{n+1} E \rightarrow f_{n} E \rightarrow s_{n} E \rightarrow f_{n+1} E[1]=\Sigma^{1,0} f_{n+1} Efn+1E→fnE→snE→fn+1E[1]=Σ1,0fn+1E
An important difference from the topological case is that this is a filtration by triangulated subcategories; the P 1 P 1 P^(1)\mathbb{P}^{1}P1-suspension is not the shift in the triangulated structure on S H ( B ) S H ( B ) SH(B)\mathrm{SH}(B)SH(B), and so the slice filtration does not arise from a t t ttt-structure.
We concentrate for a while on the case B = Spec k , k B = Spec ⁡ k , k B=Spec k,kB=\operatorname{Spec} k, kB=Spec⁡k,k a perfect field. There is an Eilenberg-MacLane functor
E M : DM ( k ) S H ( k ) E M : DM ⁡ ( k ) → S H ( k ) EM:DM(k)rarrSH(k)\mathrm{EM}: \operatorname{DM}(k) \rightarrow \mathrm{SH}(k)EM:DM⁡(k)→SH(k)
giving the motivic cohomology spectrum EM ( Z ( 0 ) ) S H ( k ) EM ⁡ ( Z ( 0 ) ) ∈ S H ( k ) EM(Z(0))inSH(k)\operatorname{EM}(\mathbb{Z}(0)) \in \mathrm{SH}(k)EM⁡(Z(0))∈SH(k) representing motivic cohomology as
H p ( X , Z ( q ) ) = EM ( Z ( 0 ) ) p , q ( X ) H p ( X , Z ( q ) ) = EM ⁡ ( Z ( 0 ) ) p , q ( X ) H^(p)(X,Z(q))=EM(Z(0))^(p,q)(X)H^{p}(X, \mathbb{Z}(q))=\operatorname{EM}(\mathbb{Z}(0))^{p, q}(X)Hp(X,Z(q))=EM⁡(Z(0))p,q(X)
One has the beautiful internal description of motivic cohomology via Voevodsky's isomorphism [122]
(2.2) s 0 S k EM ( Z ( 0 ) ) (2.2) s 0 S k ≅ EM ⁡ ( Z ( 0 ) ) {:(2.2)s_(0)S_(k)~=EM(Z(0)):}\begin{equation*} s_{0} \mathbb{S}_{k} \cong \operatorname{EM}(\mathbb{Z}(0)) \tag{2.2} \end{equation*}(2.2)s0Sk≅EM⁡(Z(0))
see also [85, THEOREM 10.5.1] and the recent paper of Bachmann-Elmanto [9]. In other words, the 0th slice truncation of the motivic sphere spectrum represents motivic cohomology. Röndigs-Østvær [103] show that the homotopy category of EM ( Z ( 0 ) ) EM ⁡ ( Z ( 0 ) ) EM(Z(0))\operatorname{EM}(\mathbb{Z}(0))EM⁡(Z(0))-modules in S H ( k ) S H ( k ) SH(k)\mathrm{SH}(k)SH(k) is equivalent to D M ( k ) D M ( k ) DM(k)\mathrm{DM}(k)DM(k) and represents the Eilenberg-MacLane functor as the forgetful functor, right-adjoint to the free EM ( Z ( 0 ) ) EM ⁡ ( Z ( 0 ) ) EM(Z(0))\operatorname{EM}(\mathbb{Z}(0))EM⁡(Z(0)) functor
DM ( k ) DM ⁡ ( k ) DM(k)\operatorname{DM}(k)DM⁡(k)
This is the triangulated motivic analog of the classical result, that the heart of the t t ttt-structure on S H S H SH\mathrm{SH}SH is A b A b Ab\mathbf{A b}Ab.

2.6. Motivic cohomology and the rational motivic stable homotopy category

In classical homotopy theory, the Eilenberg-MacLane functor EM : D ( A b ) S H D ( A b ) → S H D(Ab)rarrSHD(\mathbf{A b}) \rightarrow \mathrm{SH}D(Ab)→SH has a nice structural property: after Q Q Q\mathbb{Q}Q-localization, the functor E M Q : D ( A b ) Q S H Q E M Q : D ( A b ) Q → S H Q EM_(Q):D(Ab)_(Q)rarrSH_(Q)\mathrm{EM}_{\mathbb{Q}}: D(\mathbf{A b})_{\mathbb{Q}} \rightarrow \mathrm{SH}_{\mathbb{Q}}EMQ:D(Ab)Q→SHQ is an equivalence. Does the same happen for the motivic Eilenberg-MacLane functor E M : D M ( k ) S H ( k ) E M : D M ( k ) → S H ( k ) EM:DM(k)rarrSH(k)\mathrm{EM}: \mathrm{DM}(k) \rightarrow \mathrm{SH}(k)EM:DM(k)→SH(k) ? In general, the answer is no, and the reason goes back to Morel's C R C − R C-R\mathbb{C}-\mathbb{R}C−R dichotomy for S H ( k ) S H ( k ) SH(k)\mathrm{SH}(k)SH(k).
We discuss the case of a characteristic zero field k k kkk as base. Suppose that k k kkk admits a real embedding σ : k R σ : k → R sigma:k rarrR\sigma: k \rightarrow \mathbb{R}σ:k→R. The embedding σ σ sigma\sigmaσ induces a realization functor
R R σ : S H ( k ) S H R R σ : S H ( k ) → S H R_(R)^(sigma):SH(k)rarrSH\mathfrak{R}_{\mathbb{R}}^{\sigma}: \mathrm{SH}(k) \rightarrow \mathrm{SH}RRσ:SH(k)→SH
which sends the P 1 P 1 P^(1)\mathbb{P}^{1}P1-suspension spectrum Σ P 1 X + Σ P 1 ∞ X + Sigma_(P^(1))^(oo)X_(+)\Sigma_{\mathbb{P}^{1}}^{\infty} X_{+}ΣP1∞X+of a smooth k k kkk-scheme X X XXX to the infinite suspension spectrum of the real manifold of real points X ( R ) X ( R ) X(R)X(\mathbb{R})X(R). For an embedding σ : k C σ : k → C sigma:k rarrC\sigma: k \rightarrow \mathbb{C}σ:k→C, one has the realization functor R C σ : S H ( k ) S H R C σ : S H ( k ) → S H R_(C)^(sigma):SH(k)rarrSH\mathfrak{R}_{\mathbb{C}}^{\sigma}: \mathrm{SH}(k) \rightarrow \mathrm{SH}RCσ:SH(k)→SH, sending Σ P 1 X + Σ P 1 ∞ X + Sigma_(P^(1))^(oo)X_(+)\Sigma_{\mathbb{P}^{1}}^{\infty} X_{+}ΣP1∞X+to Σ X ( C ) + Σ ∞ X ( C ) + Sigma^(oo)X(C)_(+)\Sigma^{\infty} X(\mathbb{C})_{+}Σ∞X(C)+. If we take X = P 1 X = P 1 X=P^(1)X=\mathbb{P}^{1}X=P1, the real embedding gives you Σ S Σ S SigmaS\Sigma \mathbb{S}ΣS and the complex embedding yields Σ 2 S Σ 2 S Sigma^(2)S\Sigma^{2} \mathbb{S}Σ2S, since P 1 ( R ) = S 1 , P 1 ( C ) = S 2 P 1 ( R ) = S 1 , P 1 ( C ) = S 2 P^(1)(R)=S^(1),P^(1)(C)=S^(2)\mathbb{P}^{1}(\mathbb{R})=S^{1}, \mathbb{P}^{1}(\mathbb{C})=S^{2}P1(R)=S1,P1(C)=S2. This has the effect that the switch map τ : P 1 P 1 P 1 P 1 Ï„ : P 1 ∧ P 1 → P 1 ∧ P 1 tau:P^(1)^^P^(1)rarrP^(1)^^P^(1)\tau: \mathbb{P}^{1} \wedge \mathbb{P}^{1} \rightarrow \mathbb{P}^{1} \wedge \mathbb{P}^{1}Ï„:P1∧P1→P1∧P1 induces an automorphism of S k S k S_(k)\mathbb{S}_{k}Sk that maps to -1 under the real embedding and to +1 under the complex embedding. Thus, if we invert 2 and decompose the motivic sphere spectrum into ± 1 ± 1 +-1\pm 1±1 eigenfactors with respect to τ Ï„ tau\tauÏ„, we decompose S H ( k ) [ 1 / 2 ] S H ( k ) [ 1 / 2 ] SH(k)[1//2]\mathrm{SH}(k)[1 / 2]SH(k)[1/2] into corresponding summands S H ( k ) ± S H ( k ) ± SH(k)_(+-)\mathrm{SH}(k)_{ \pm}SH(k)±, with all of S H ( k ) + S H ( k ) + SH(k)_(+)\mathrm{SH}(k)_{+}SH(k)+going to zero under the real embedding and all of
Alternatively, the minus part is SH ( k ) [ 1 / 2 , η 1 ] SH ⁡ ( k ) 1 / 2 , η − 1 SH(k)[1//2,eta^(-1)]\operatorname{SH}(k)\left[1 / 2, \eta^{-1}\right]SH⁡(k)[1/2,η−1], where η η eta\etaη is the P 1 P 1 P^(1)\mathbb{P}^{1}P1-stabilization of the algebraic Hopf map
η : A 2 { 0 } P 1 , η ( x , y ) = [ x : y ] η : A 2 ∖ { 0 } → P 1 , η ( x , y ) = [ x : y ] eta:A^(2)\\{0}rarrP^(1),quad eta(x,y)=[x:y]\eta: \mathbb{A}^{2} \backslash\{0\} \rightarrow \mathbb{P}^{1}, \quad \eta(x, y)=[x: y]η:A2∖{0}→P1,η(x,y)=[x:y]
A motivic spectrum E S H ( k ) E ∈ S H ( k ) E inSH(k)E \in \mathrm{SH}(k)E∈SH(k) is orientable if E E EEE has a good theory of Thom classes. For V X V → X V rarr XV \rightarrow XV→X a vector bundle with 0 -section s 0 : X V s 0 : X → V s_(0):X rarr Vs_{0}: X \rightarrow Vs0:X→V, we have the Thom space Th ( V ) := V / ( V s 0 ( X ) ) H ( k ) Th ⁡ ( V ) := V / V ∖ s 0 ( X ) ∈ H ∙ ( k ) Th(V):=V//(V\\s_(0)(X))inH_(∙)(k)\operatorname{Th}(V):=V /\left(V \backslash s_{0}(X)\right) \in \mathscr{H}_{\bullet}(k)Th⁡(V):=V/(V∖s0(X))∈H∙(k) (defined as the quotient of representable presheaves). An orientation for E E EEE consists of giving a class
th ( V ) E 2 r , r ( Th ( V ) ) th ⁡ ( V ) ∈ E 2 r , r ( Th ⁡ ( V ) ) th(V)inE^(2r,r)(Th(V))\operatorname{th}(V) \in E^{2 r, r}(\operatorname{Th}(V))th⁡(V)∈E2r,r(Th⁡(V))
for each rank r r rrr vector bundle V X V → X V rarr XV \rightarrow XV→X over X S m k X ∈ S m k X inSm_(k)X \in \mathrm{Sm}_{k}X∈Smk, satisfying axioms parallel to the notion of a C C C\mathbb{C}C-orientation in topology; a choice of Thom classes defines E E EEE as an oriented cohomology theory. After inverting 2, all the orientable E E EEE live in the plus part; this includes motivic cohomology, as well as algebraic K K KKK-theory and algebraic cobordism. These theories E E EEE all have the property that η η eta\etaη induces zero on E E EEE-cohomology.
Theories that live in the minus part will contrariwise invert η η eta\etaη (after inverting 2); these include things like Witt theory or cohomology of the sheaf of Witt groups. The real and complex avatars of this are seen by noting that the complex realization of the algebraic Hopf map is the usual Hopf map, which is the 2-torsion element of stable π 1 Ï€ 1 pi_(1)\pi_{1}Ï€1 of the sphere spectrum, while the real realization is the multiplication map × 2 : S 1 S 1 × 2 : S 1 → S 1 xx2:S^(1)rarrS^(1)\times 2: S^{1} \rightarrow S^{1}×2:S1→S1.
The analog of the fact that E M Q : D ( Q ) S H Q E M Q : D ( Q ) → S H Q EM_(Q):D(Q)rarrSH_(Q)\mathrm{EM}_{\mathbb{Q}}: D(\mathbb{Q}) \rightarrow \mathrm{SH}_{\mathbb{Q}}EMQ:D(Q)→SHQ is an equivalence is the following result of Cisinski-Déglise
Theorem 2.4 ([33, THEOREM 16.2.13]). The unit map S k EM ( Z ( 0 ) ) S k → EM ⁡ ( Z ( 0 ) ) S_(k)rarr EM(Z(0))\mathbb{S}_{k} \rightarrow \operatorname{EM}(\mathbb{Z}(0))Sk→EM⁡(Z(0)) induces an isomorphism
S H ( k ) + Q DM ( k ) Q S H ( k ) + Q → DM ⁡ ( k ) Q SH(k)_(+Q)rarr DM(k)_(Q)\mathrm{SH}(k)_{+\mathbb{Q}} \rightarrow \operatorname{DM}(k)_{\mathbb{Q}}SH(k)+Q→DM⁡(k)Q
with inverse the Eilenberg-MacLane functor followed by the plus-projection
D M ( k ) Q S H ( k ) Q S H ( k ) + Q D M ( k ) Q → S H ( k ) Q → S H ( k ) + Q DM(k)_(Q)rarrSH(k)_(Q)rarrSH(k)_(+Q)\mathrm{DM}(k)_{\mathbb{Q}} \rightarrow \mathrm{SH}(k)_{\mathbb{Q}} \rightarrow \mathrm{SH}(k)_{+\mathbb{Q}}DM(k)Q→SH(k)Q→SH(k)+Q
The rational minus part is also a homotopy category of modules over a suitable cohomology theory, namely Witt sheaf cohomology. For a field F F FFF, we have the Witt ring W ( F ) W ( F ) W(F)W(F)W(F) of virtual non-degenerate quadratic forms, modulo the hyperbolic form. This extends to a sheaf W W W\mathcal{W}W on S m k S m k Sm_(k)\mathrm{Sm}_{k}Smk, and the functor X H N i s p ( X , W ) X ↦ H N i s p ( X , W ) X|->H_(Nis)^(p)(X,W)X \mapsto H_{\mathrm{Nis}}^{p}(X, \mathcal{W})X↦HNisp(X,W) is represented in S H ( k ) S H ( k ) SH(k)\mathrm{SH}(k)SH(k) by a suitable spectrum EM ( W ) EM ⁡ ( W ) EM(W)\operatorname{EM}(\mathcal{W})EM⁡(W). We have
Theorem 2.5 ([3, THEOREM 4.2, COROLLARY 4.4]). The functor E E M ( W ) Q E E ↦ E M ( W ) Q ∧ E E|->EM(W)_(Q)^^EE \mapsto \mathrm{EM}(\mathcal{W})_{\mathbb{Q}} \wedge EE↦EM(W)Q∧E induces a natural isomorphism of S H ( k ) Q S H ( k ) − Q SH(k)_(-Q)\mathrm{SH}(k)_{-\mathbb{Q}}SH(k)−Q with the homotopy category E M ( W ) Q E M ( W ) Q EM(W)_(Q)\mathrm{EM}(\mathcal{W})_{\mathbb{Q}}EM(W)Q-modules.
From this point of view, one can see the Z Z Z\mathbb{Z}Z-graded cohomology theory
X n 0 H N i s n ( X , W ) X ↦ ⨁ n ≥ 0   H N i s n ( X , W ) X|->bigoplus_(n >= 0)H_(Nis)^(n)(X,W)X \mapsto \bigoplus_{n \geq 0} H_{\mathrm{Nis}}^{n}(X, \mathcal{W})X↦⨁n≥0HNisn(X,W)
as the motivic cohomology for the minus part; this theory picks up information about the real points of schemes. To get the complete theory, one also needs to include twists of W W W\mathcal{W}W by line bundles, an analog of orientation local systems in the topological setting. We will say more about this in Section 4.

2.7. Slice tower and motivic Atiyah-Hirzebruch spectral sequences

The classical Atiyah-Hirzebruch spectral sequence for a spectrum E S H E ∈ S H E inSHE \in \mathrm{SH}E∈SH is the spectral sequence of the Postnikov tower of E E EEE, and looks like
E 2 p , q := H p ( X , π q E ) E p + q ( X ) E 2 p , q := H p X , Ï€ − q E ⇒ E p + q ( X ) E_(2)^(p,q):=H^(p)(X,pi_(-q)E)=>E^(p+q)(X)E_{2}^{p, q}:=H^{p}\left(X, \pi_{-q} E\right) \Rightarrow E^{p+q}(X)E2p,q:=Hp(X,π−qE)⇒Ep+q(X)
This comes by identifying the q q qqq th layer in the Postnikov tower with the shifted EilenbergMacLane spectrum Σ q E M ( π q ( E ) ) Σ q E M Ï€ q ( E ) Sigma^(q)EM(pi_(q)(E))\Sigma^{q} \mathrm{EM}\left(\pi_{q}(E)\right)ΣqEM(Ï€q(E)).
Together with results of Pelaez [99] and Gutierrez-Röndigs-Spitzweck, Voevodsky's isomorphism (2.2) has a structural expression, namely, for any E S H ( k ) E ∈ S H ( k ) E inSH(k)E \in \mathrm{SH}(k)E∈SH(k), each slice s q ( E ) s q ( E ) s_(q)(E)s_{q}(E)sq(E) has a canonical structure of an EM ( Z ( 0 ) ) EM ⁡ ( Z ( 0 ) ) EM(Z(0))\operatorname{EM}(\mathbb{Z}(0))EM⁡(Z(0))-module. We write corresponding object of DM ( k ) DM ⁡ ( k ) DM(k)\operatorname{DM}(k)DM⁡(k) as π q mot ( E ) Ï€ q mot ( E ) pi_(q)^(mot)(E)\pi_{q}^{\operatorname{mot}}(E)Ï€qmot(E), satisfying
s q ( E ) = Σ P 1 q EM ( π q mot ( E ) ) = S 2 q , q EM ( π q mot ( E ) ) s q ( E ) = Σ P 1 q EM ⁡ Ï€ q mot ( E ) = S 2 q , q ∧ EM ⁡ Ï€ q mot ( E ) s_(q)(E)=Sigma_(P^(1))^(q)EM(pi_(q)^(mot)(E))=S^(2q,q)^^EM(pi_(q)^(mot)(E))s_{q}(E)=\Sigma_{\mathbb{P}^{1}}^{q} \operatorname{EM}\left(\pi_{q}^{\operatorname{mot}}(E)\right)=S^{2 q, q} \wedge \operatorname{EM}\left(\pi_{q}^{\operatorname{mot}}(E)\right)sq(E)=ΣP1qEM⁡(Ï€qmot(E))=S2q,q∧EM⁡(Ï€qmot(E))
This gives the motivic Atiyah-Hirzebruch spectral sequence
E 2 p , q ( n ) := H p q ( X , π q m o t ( E ) ( n q ) ) E p + q , n ( X ) E 2 p , q ( n ) := H p − q X , Ï€ − q m o t ( E ) ( n − q ) ⇒ E p + q , n ( X ) E_(2)^(p,q)(n):=H^(p-q)(X,pi_(-q)^(mot)(E)(n-q))=>E^(p+q,n)(X)E_{2}^{p, q}(n):=H^{p-q}\left(X, \pi_{-q}^{\mathrm{mot}}(E)(n-q)\right) \Rightarrow E^{p+q, n}(X)E2p,q(n):=Hp−q(X,π−qmot(E)(n−q))⇒Ep+q,n(X)
These slices have been explicitly identified in a number of important cases. The first case was algebraic K K KKK-theory, K G L S H ( k ) K G L ∈ S H ( k ) KGLinSH(k)\mathrm{KGL} \in \mathrm{SH}(k)KGL∈SH(k). Voevodsky [118,119] and Levine [85] show
s q ( K G L ) = EM ( Z ( q ) [ 2 q ] ) = Σ P 1 q EM ( Z ( 0 ) ) s q ( K G L ) = EM ⁡ ( Z ( q ) [ 2 q ] ) = Σ P 1 q EM ⁡ ( Z ( 0 ) ) s_(q)(KGL)=EM(Z(q)[2q])=Sigma_(P^(1))^(q)EM(Z(0))s_{q}(\mathrm{KGL})=\operatorname{EM}(\mathbb{Z}(q)[2 q])=\Sigma_{\mathbb{P}^{1}}^{q} \operatorname{EM}(\mathbb{Z}(0))sq(KGL)=EM⁡(Z(q)[2q])=ΣP1qEM⁡(Z(0))
so
π q m o t ( K G L ) = Z ( 0 ) Ï€ q m o t ( K G L ) = Z ( 0 ) pi_(q)^(mot)(KGL)=Z(0)\pi_{q}^{\mathrm{mot}}(\mathrm{KGL})=\mathbb{Z}(0)Ï€qmot(KGL)=Z(0)
corresponding to classical computation for topological K K KKK-theory,
π q s K U = { Z for q even 0 for q odd. Ï€ q s K U = Z  for  q  even  0  for  q  odd.  pi_(q)^(s)KU={[Z," for "q" even "],[0," for "q" odd. "]:}\pi_{q}^{s} K U= \begin{cases}\mathbb{Z} & \text { for } q \text { even } \\ 0 & \text { for } q \text { odd. }\end{cases}Ï€qsKU={Z for q even 0 for q odd. 
Using "algebraic Bott periodicity" for KGL: K G L a + 2 n , b + n ( X ) = K G L a , b ( X ) = K 2 b a ( X ) K G L a + 2 n , b + n ( X ) = K G L a , b ( X ) = K 2 b − a ( X ) KGL^(a+2n,b+n)(X)=KGL^(a,b)(X)=K_(2b-a)(X)\mathrm{KGL}^{a+2 n, b+n}(X)=\mathrm{KGL}^{a, b}(X)=K_{2 b-a}(X)KGLa+2n,b+n(X)=KGLa,b(X)=K2b−a(X), this yields the Atiyah-Hirzebruch spectral sequence of the Beilinson-Lichtenbaum axiom (v),
E 2 p , q := H p q ( X , Z ( q ) ) K p q ( X ) E 2 p , q := H p − q ( X , Z ( − q ) ) ⇒ K − p − q ( X ) E_(2)^(p,q):=H^(p-q)(X,Z(-q))=>K_(-p-q)(X)E_{2}^{p, q}:=H^{p-q}(X, \mathbb{Z}(-q)) \Rightarrow K_{-p-q}(X)E2p,q:=Hp−q(X,Z(−q))⇒K−p−q(X)
There is also a corresponding spectral sequence with Z / m Z / m Z//m\mathbb{Z} / mZ/m-coefficients.
This Atiyah-Hirzebruch spectral sequence for algebraic K K KKK-theory was first constructed for X X XXX the spectrum of a field by Bloch and Lichtenbaum [29], by a completely different approach and without recourse to motivic homotopy theory or Voevodsky's slice tower. Their construction was generalized to general X S m k X ∈ S m k X inSm_(k)X \in \mathrm{Sm}_{k}X∈Smk by Friedlander-Suslin [43], also without using the categorical machinery. The rough idea is to give a filtration by codimension of support on X × Δ X × Δ ∗ X xxDelta^(**)X \times \Delta^{*}X×Δ∗ (with additional conditions), and then identify the layers with a suitable complex of cycles. Another approach, by Grayson [54], relies on the K K KKK-theory of exact categories with commuting isomorphisms. For smooth finite-type schemes over a perfect field, all these approaches yield the same spectral sequence (see [85, THEOREM 7.1.1, THEOREM 9.0.3], [44])).

3. MOTIVIC COHOMOLOGY OVER A GENERAL BASE

It is natural to ask if this picture of a good motivic cohomology theory for schemes over a perfect field can be extended to more general base-schemes, not just as an interesting technical question but for a wide range of applications, especially in arithmetic. Over a perfect field, we have a number of different constructions that all lead to the same groups, each of which have their advantages and disadvantages: Bloch's higher Chow groups, the cohomology of a suitable Suslin complex, the morphisms in D M ( k ) D M ( k ) DM(k)\mathrm{DM}(k)DM(k), the cohomology theory represented in S H ( k ) S H ( k ) SH(k)\mathrm{SH}(k)SH(k) by EM ( Z ( 0 ) ) EM ⁡ ( Z ( 0 ) ) EM(Z(0))\operatorname{EM}(\mathbb{Z}(0))EM⁡(Z(0)), or by s 0 S k s 0 S k s_(0)S_(k)s_{0} \mathbb{S}_{k}s0Sk, or by s 0 K G L s 0 K G L s_(0)KGLs_{0} \mathrm{KGL}s0KGL.
One would expect motivic cohomology to be an absolute theory, like algebraic K K − K-K-K− theory, that is, its value on a given scheme should not depend on the choice of base-scheme. In terms of a spectrum H Z S S H ( S ) H Z S ∈ S H ( S ) HZ_(S)inSH(S)H \mathbb{Z}_{S} \in \mathrm{SH}(S)HZS∈SH(S) that would represent our putative theory, this is the cartesian condition: there should be canonical isomorphisms H Z T f H Z S H Z T ≅ f ∗ H Z S HZ_(T)~=f^(**)HZ_(S)H \mathbb{Z}_{T} \cong f^{*} H \mathbb{Z}_{S}HZT≅f∗HZS for each morphism of schemes f : T S f : T → S f:T rarr Sf: T \rightarrow Sf:T→S.
The identity (2.2) raises the possibility of defining motivic cohomology over a general base-scheme B B BBB by this formula. One problem here is that the slice filtration has only a limited functoriality: for f : C B f : C → B f:C rarr Bf: C \rightarrow Bf:C→B a map of schemes, one does not in general have a natural isomorphism f s 0 s 0 f f ∗ ∘ s 0 ≅ s 0 ∘ f ∗ f^(**)@s_(0)~=s_(0)@f^(**)f^{*} \circ s_{0} \cong s_{0} \circ f^{*}f∗∘s0≅s0∘f∗. For the cartesian property to hold for a motivic cohomology defined via the slice filtration, one would want the compatibility of the slices
with pullback; this latter is in fact the case for f : C B f : C → B f:C rarr Bf: C \rightarrow Bf:C→B is a morphism of separated, finite type schemes over a field k k kkk of characteristic zero (or assuming resolution of singularities for separated, finite type k k kkk-schemes), by results of Pelaez [100, COROLLARY 4.3]. This compatibility also holds for arbitrary smooth f f fff, but is not known in general.
Another concrete candidate for the motivic Borel-Moore homology is given by the hypercohomology of a version of Bloch's cycle complex, suitably extended to the setting of finite type schemes over a Dedekind domain. This theory is nearly absolute, as it depends only on a good notion of dimension or codimension, which one would have for say equiKrull-dimensional schemes. In general, however, this theory lacks a full functoriality under pullback and also lacks a multiplicative structure.
There is a P 1 P 1 P^(1)\mathbb{P}^{1}P1-spectrum K G L S S H ( S ) K G L S ∈ S H ( S ) KGL_(S)inSH(S)\mathrm{KGL}_{S} \in \mathrm{SH}(S)KGLS∈SH(S) that represents the so-called homotopy invariant K K KKK-theory over an arbitrary base and is cartesian, so one could try s 0 s 0 s_(0)s_{0}s0 KGL as a representing spectrum. Again, the problem is the functoriality of the slice filtration, but perhaps KGL would be easier to handle than the sphere spectrum in this regard.

3.1. Cisinski-Déglise motivic cohomology

Over an base-scheme S S SSS that is noetherian and of finite Krull dimension, CisinskiDéglise [33, §11] have followed the program of Voevodsky to define a triangulated category of motives D M C D ( S ) D M C D ( S ) DM_(CD)(S)\mathrm{DM}_{\mathrm{CD}}(S)DMCD(S), with Tate objects Z S ( n ) Z S ( n ) Z_(S)(n)\mathbb{Z}_{S}(n)ZS(n), and with a "motives functor"
M : Sm S DM C D ( S ) ; X M ( X ) DM C D ( S ) M : Sm S → DM C D ⁡ ( S ) ; X ↦ M ( X ) ∈ DM C D ⁡ ( S ) M:Sm_(S)rarrDM_(CD)(S);X|->M(X)inDM_(CD)(S)M: \operatorname{Sm}_{S} \rightarrow \operatorname{DM}_{\mathrm{CD}}(S) ; X \mapsto M(X) \in \operatorname{DM}_{\mathrm{CD}}(S)M:SmS→DMCD⁡(S);X↦M(X)∈DMCD⁡(S)
This extends Voevodsky's construction of D M ( k ) D M ( k ) DM(k)\mathrm{DM}(k)DM(k) for a perfect field k k kk